GeoInformatica

, Volume 18, Issue 3, pp 461–500 | Cite as

TARS: traffic-aware route search

Article

Abstract

In a traffic-aware route search (TARS), the user provides start and target locations and sets of search terms. The goal is to find the fastest route from the start location to the target via geographic entities (points of interest) that correspond to the search terms, while taking into account variations in the travel speed due to changes in traffic conditions, and the possibility that some visited entities will not satisfy the search requirements. A TARS query may include temporal constraints and order constraints that restrict the order by which entities are visited. Since TARS generalizes the Traveling-Salesperson Problem, it is an NP-hard problem. Thus, it is unlikely to find a polynomial-time algorithm for evaluating TARS queries. Hence, we present in this paper three heuristics to answer TARS queries—a local greedy approach, a global greedy approach and an algorithm that computes a linear approximation to the travel speeds, formulates the problem as a Mixed Integer Linear Programming (MILP) problem and uses a solver to find a solution. We provide an experimental evaluation based on actual traffic data and show that using a MILP solver to find a solution is effective and can be done within a limited running time in many real-life scenarios. The local-greedy approach is the least effective in finding a fast route, however, it has the best running time and it is the most scalable.

Keywords

Geographic information systems Route search Temporal constraints Probabilistic data  Heuristic algorithms Traffic 

1 Introduction

Geographical search is a fundamental part of the World-Wide Web, e.g., Bing Maps, Google Maps and Yahoo! Local are popular geographical search engines. Recently, geographical applications have also become ubiquitous by being prevalently available on hand-held devices, such as smart phones, PDAs, and car navigation systems. The commonly-used geographic search applications receive keywords and depict relevant points of interest on a map or find a route between two specified addresses. Points of interest represent different real-world geographical entities such as buildings, shops, train stations, parks, tourist attractions, etc.

An important difference between geographical search and ordinary non-geographical keyword search is that in a geographical search users frequently conduct the search with an intention to actually visit the geographical entities of the result. Thus, in many scenarios the result of a geographical search should be a route, and it may be desired that the route will go via several points of interest that represent different types of geographical entities. The task of formulating such search query and answering it is called route search.

In a route search, the user specifies a start location, a target destination and a set of geographical search subqueries. A typical search query comprises a set of keywords where the keywords specifies the type of geographical entities the user wants to visit. The goal is to find a route that goes from the start location to the target location via geographical entities that are returned by the search subqueries. The following example illustrates such a route search.

Example 1

A businessperson, Alice, has an important out-of-town meeting. Prior to the meeting, she needs to find a computer store for replacing the failing battery of her laptop computer. In addition, she needs to go via a gas station to fuel up her car, and she wants to have lunch in a vegetarian restaurant, either before or after the meeting. Searching for the relevant entities using an ordinary geographical search engine and planning an effective route via the entities is a difficult task. Taking into account traffic conditions and temporal constraints, such as the start time of the meeting, increases the intricacy of the problem. All this also needs to be done under conditions of uncertainty where some computer stores may not have a suitable battery for the specific model of Alice’s laptop, yet this will only be discovered upon arrival at these stores. Thus, the route may need to go via several computer stores—not too many so that the route will not be longer than necessary, and not too few so that with high probability Alice will find an appropriate battery.

Dealing with the uncertainty caused by the possibility that entities will fail to satisfy the user increases the complexity of computing a route even further. The difficulty is because users may discover whether an entity satisfies the search requirements only upon arrival at the entity. For instance, in Example 1 Alice will know for certain whether a computer store has in its stock a suitable battery for her laptop only upon arrival at the store.

Based on statistical analysis of historical queries, the probability of satisfying the user can be assigned to each entity returned by a subquery. We refer to a set of entities with their assigned success probabilities as a probabilistic dataset. In a probabilistic dataset, a probability value is attached to each represented entity. This value specifies the likelihood of the entity to satisfy the user with respect to the corresponding subquery. The probabilities must be considered when computing a route. First, probabilities define a preference relationship where an entity with high probability is preferred to an entity with low probability. Secondly, when constructing a route, it is important to have a recovery plan, so that if certain entities along the route fail to satisfy the user, the user can visit more entities of the same type without increasing the travel time more than necessary. Several papers have dealt with route search over non-probabilistic datasets (see [17, 32, 39]) and other investigated route search over probabilistic datasets (see [9, 23, 24, 25, 37]). Some papers addressed the problem of dealing with partial order constraints [17, 24]. However, they did not handle route search in the presence of traffic and temporal constraints.

To deal with tasks such as the search in Example 1, subqueries should include order constraints and temporal constraints. An order constraint specifies that entities of some type should only be visited after visiting an entity of some other type. In Example 1, Alice should go by a computer store before reaching the meeting place.

Temporal constraints specify limitations on the time during the day when entities should be arrived at. In Example 1, a temporal constraint would specify that Alice should arrive at the place of the meeting before the meeting starts. Another constraint may limit the visit at the restaurant to be around noon, so that lunch would not be too early or too late.

In addition to constraints that are part of the query, there are temporal constraints in the dataset. For instance, institutions, such as a museum, have opening hours. The user should arrive at a museum during the opening hours. However, reaching a museum five minutes before closing time is pointless. Thus, the route should be computed while taking into account an estimated stay duration in the entities. Estimated stay durations are also required for determining the departure time, and the departure time affects the time it takes to travel to the next entity (because travel times are inconstant).

An answer to a route-search query is a route that travels via entities of all the specified types (e.g., in Example 1 the route should go via a gas station, computer stores, a restaurant and the meeting place) where the times of the visits should adhere to the order and temporal constraints. Note that in this context, a route must also include a departure time since it affects the travel duration and the satisfaction of the temporal constraints.

When calculating the travel time of a route and planning the route so that it will adhere to the temporal constraints of the query, it is necessary to effectively model travel durations. Unlike distances, travel durations are variable. In many cities, the travel speed on major arteries during rush hours is significantly slower than during other hours, yet, it is difficult to calculate the effect of the traffic load, because road networks are unevenly affected by congestion [19]. Previous papers on route search have focused on finding the shortest route. They did not address the issue of varying traffic conditions and handling time constraints, although finding the fastest route while satisfying temporal constraints is required in many real-world scenarios. In this paper, the goal is to find the fastest route along with an optimal departure time, while considering traffic conditions and temporal constraints.

Note that merely finding a route that visits the entities within their temporal constraint is a constraint-satisfaction problem. Such problems may not have a solution at all or may have multiple solutions. Generally, constraint satisfaction problems are NP-complete [7]. Hence, a route-search problem with temporal constraints, even when assuming constant travel speeds (i.e., assuming constant traffic conditions), is both a hard routing problem and a hard scheduling problem. We refer to a route search in the presence of varying traffic conditions, temporal constraints, and order constrains as Traffic-Aware Route Search (TARS). This paper formally models the problem and presents three heuristic algorithms for answering TARS queries while striving to minimize the overall travel time. We tested and compared the proposed algorithms using real road networks and actual traffic data.

Route search can be performed as an interactive process. The interactive (or online) approach (see [13, 23, 24]) is designed for users that apply the search using a device such as a smartphone. In such search the route is computed in real time, (i.e., while users are traveling). The smartphone allows the user to provide feedbacks to the systems and it can present the result of modifications in the route, thus it supports construction of routes interactively. The non-interactive approach has the form of a pre-planned search (see [9, 17, 25, 32, 37, 39]), and it is designed for setting in which planning is required, e.g., for determining an optimal departure time. This approach is also useful in cases where the search is conducted using a device that has a limited network connectivity or no GPS receiver. In this setting the goal is to plan an optimal pre-calculated route that will take into account possible failures, of visited entities, to satisfy the user. The two approaches both deal with different cases and are hence complementary methods. In this paper we deal with the pre-planned setting.

This paper is organized as follows. Section 2 illustrates how users can formulate TARS queries using a simple user interface which is part of a system we have developed to pose and answer TARS queries. In Section 3, we present our framework and formally define TARS. Section 4 presents three algorithms for answering TARS queries. These algorithms take into account order constraints, temporal constraints and varying traffic conditions. In Section 5, we present an experimental evaluation of the algorithms. We compare the effectiveness of the algorithms in finding a fast route and their running time efficiency. Section 6 surveys related work. Finally, in Section 7, we conclude.

2 TARS queries

In this section we illustrate the formulation of TARS queries using a graphical user interface. The purpose of this section to show that while TARS queries have a complex definition, as they comprise different types of constraints, their formulation can be simple. We do not try to provide here a comprehensive study of user interfaces for route search.

Systems that support TARS queries can either be designed to serve experts in the context of specific domains or can be intended for laymen users by providing a suitable user interface. Examples of domain-specific applications are planning guided tours, planning routes for interviewers to efficiently conduct face-to-face surveys, planning routes for scientific exploration missions, finding fast routes for rescue vehicles such as ambulances, etc. Such tasks are typically done by professionals. To be suitable for laymen users, the task of posing a query should remain simple, while the complexities involved in answering it stay hidden behind the scenes.

To conduct our research, we developed a system that answers TARS queries and allows users to easily formulate them [18]. The formulation of a TARS query involves three simple steps. First, the user sets the origin and the destination. Secondly, the user specifies the types of entities (stops) that should be visited along the route. Thirdly, temporal and order constraints can be added.

Figure 1 illustrates a formulation of a TARS query using the system. The user begins by providing North Beach, San Fransisco as the origin address (the start location can also be obtained as a GPS measure), followed by Sunset District, San Fransisco as the destination address (the target). The user proceeds by adding the search terms, (1) coffee, (2) ATM, (3) shoe store and (4) vegetarian restaurant as her desired stops. Next, the user can optionally set the temporal constraints for the origin, destination and stops. The temporal constraints consist of earliest and latest arrival times and, for stops, an estimated stay duration. There is an option to set an importance degree (priority), for each stop. This value indicates the importance of visiting a satisfying entity of the corresponding type. Increasing it will typically cause the resulting route to go via more entities of that type so that if one entity fails to satisfy the user, there are alternative entities of the same type along the route to the target destination. Finally, the user expresses a requirement to visit an ATM before arriving at the shoe store. This is done by adding an order constraint on these two stops.
Fig. 1

An illustration of a TARS query formulated using the graphical user interface. The query is posed over an area in the city of San Fransisco. The icons depicted on the map represent spatial entities that match the given search terms. Stops represent the entities that should be visited. The columns shown for the stops are as follows: (1) “Stop name” presents the search terms that define the type of the entity, (2) the symbol “#” indicates the maximal number of required search results, (3) “Start” and “End” refer to the earliest and latest allowed arrival times, (4) “Mins” is the estimated duration at the stops, and (5) “Priority” indicates the importance of visiting a satisfying entity. Under “Constraints”, there is an order constraint to specify that an ATM should be visited before any shoe store

TARS is useful not only for complex route planning, it can also facilitate ordinary daily tasks. As an example, consider a user who needs to buy groceries on the way home from work. Current route search approaches can find for the user the shortest route home via a grocery store. However, the provided route would not indicate the optimal departure time and it will not necessarily be the fastest route for a specific departure time. Formulating this task as a TARS query will allow the user to find an effective route while taking into account limitations on the departure time and the traffic in the area.

3 Framework

In this section, we present our framework, we formally define the concept of Traffic-Aware Route Search (TARS), and we explain how order constraints, temporal constraints and traffic conditions are modeled. The model is designed to include all the route search aspects that have been considered in previous papers to provide a comprehensive solution.

Dataset

A geospatial dataset is a collection of geospatial objects. Each object represents a real-world geographical entity, and its location is the same as that of the entity. An object may have additional spatial attributes, such as height or shape, and non-spatial attributes, such as type or name. We assume that locations are points. Thus, for objects that are represented by a polygonal shape and do not have a specified point location, an arbitrary point inside them is chosen to be the point location. Generally “object” and “entity” are considered synonyms, however in our terminology, an object is a representation of a real-world entity.

An object may have an opening time and a closing time, which represent the time during the day when the entity is available. For example, a museum opening times may be from 10:00 till 18:00. To simplify the model we only consider cases where the opening times are continuous. Cases where opening times are discontinuous can easily be solved by cloning objects and assigning to each clone a continuous fragment of the opening times. Similarly, cases where opening hours change from one day to another can be solved by cloning objects so that each clone will represent the opening hours on a different day, and then use the appropriate clone according to the day of the travel. For objects that are open 24 h a day, we consider the opening time to be 00:00 and the closing time to be 23:59. For an object o′, we denote by \(t_o(o^{\prime})\) and \(t_c(o^{\prime})\) the opening time and closing time of o′.

Search subqueries

Users specify the entities they wish to visit using search subqueries (subqueries, for short). A search subquery contains a set of keywords, and it may include different constraints on the spatial and non-spatial attributes of objects. The result of a subquery is represented as a probabilistic dataset, where each object is assigned a value 0 ≤ p ≤ 1, called probability of success (or probability, for short). The probability of an object o indicates the likelihood that the entity represented by o actually satisfies the requirements of the user. For example, a restaurant called “Pizza House” is more likely to satisfy a search for a vegetarian restaurant than a place called “Steak House”. Assigning probabilities to objects can be done using a combination of information-retrieval techniques and statistical analysis of historical user feedback data. However, how to do so is beyond the scope of this paper.

A user may need to visit several entities of the same type to satisfy a search subquery. This is because the dataset is probabilistic, and because in many cases only upon arrival at the entity the user knows whether the subquery is satisfied. For example, before finding a pair of satisfactory shoes, the user may visit several shoe stores that are too expensive or that are not his style. Thus, when planning a route, the visited entities are determined before the travel starts, and only the probability of success is known a priori.

Note that in our context, whether a visited object o satisfies a user is either true or false. It merely indicates whether the user wants to visit additional objects of the same type as o. For instance, after buying a laptop battery, Alice (from Example 1) will not visit additional computer stores.

In a temporal search subquery, there are three types of temporal constraints. The earliest arrival time and latest arrival time specify lower and upper bounds on the arrival time. For example, a user may specify that she wishes to arrive at a coffee shop no earlier than 9:00 and no later than 11:00. The estimated duration time specifies the expected length of the stay at the entity. For example, a user may estimate a stay duration of an hour at a shopping mall. We denote a temporal subquery as a four-tuple Q = (q, e, l, d) where q is a set of keywords, e and l are the earliest and latest arrival times, and d is the estimated duration time.

TARS queries

A user specifies her search requirements in the form of a TARS query. A query comprises start and end locations, denoted s and t, a set \(\mathcal{Q}\) of temporal search subqueries to define the types of entities that the route should visit, and a set O of order constraints that define the order by which entities should be visited.

Formally, a TARS query is a four-tuple \(T=(\bar s, \bar t, \mathcal{Q}, O)\). The start point\(\bar s=(s, [e_s,l_s])\) defines that the route should start in the location s, where the departure time from s should not be before es or after ls. The target point\(\bar t=(t, [e_{t},l_{t}])\) defines a destination location t, such that the arrival at t should be not before et or after lt.

The set \(\mathcal{Q}=\{(Q_1,\tau_1)\ldots,(Q_m,\tau_m)\}\) defines m temporal subqueries of the form Qi = (qi, ei, li, di), where qi is a set of keywords, ei is the earliest allowed arrival time, li is the latest allowed arrival time and di denotes the estimated stay duration. Each subquery has a corresponding probability thresholdτi. The probability threshold τi of Qi requires the following: The probability that at least one visited object will satisfy Qi should not be smaller than τi. (In the system that was illustrated in Section 2, the threshold is computed based on the importance degree of the subquery.) For instance, consider a subquery Qi with search terms “French restaurant” and a probability threshold τi = 0.8. There is a chance that the first visited entity will not satisfy the user, e.g., there are no available tables at the restaurant. Thus, the route should go via several objects that satisfy the search subquery, i.e., through several French restaurants. The probability threshold τi specifies the importance of visiting a French restaurant, and hence, it affects the number of restaurants on the route. In this case, the chances of satisfying the user should be at least 80 percent. Accordingly, the algorithms may prefer clusters of relevant entities on isolated entities. If, when traversing the route, the user is satisfied with one of the restaurants, she can skip the following restaurants along the route.

Finally, \(O\subset \mathcal{Q}\times \mathcal{Q}\) is a set of pairs (Qi, Qj) that specify order constraints on the search subqueries. A pair (Qi, Qj) specifies an order where objects that satisfy Qj should only be visited after visiting the objects that satisfy Qi. The notations are listed in Table 1.
Table 1

Notations of a TARS query

Notations

Description

\(T=(\bar s, \bar t, \mathcal{Q}, O)\)

TARS query represented by a 4-tuple

\(\bar s=(s, [e_s,l_s])\)

Location s denotes the start location. The interval specifies that the departure time from s should not be before es or after ls

\(\bar t=(t, [e_t,l_t])\)

Location t denotes the target location. The interval denotes that the arrival time at t should not be before et or after lt

\(\mathcal{Q}=\{(Q_1,\tau_1)\ldots,(Q_m,\tau_m)\}\)

An m-tuple defining m temporal search subqueries and a minimum probability thresholds for each subquery

Qi = (qi, ei, li, di)

A search subquery, where qi is a set of keywords, ei and li denote the earliest and latest allowed arrival times, respectively, and di denotes the estimated stay duration

p

Probability of success of an object in O, where 0 ≤ p ≤ 1

\(O\subset \mathcal{Q}\times \mathcal{Q}\)

A set of order constraints

Road network

TARS queries are posed over a road network. A road network is a directed graph G = (V,E) whose nodes V represent junctions and the edges E represent roads. Each junction has a point location and a road is represented by a polygonal line that connects two junctions. In the presence of traffic, the travel time on each road depends on the traffic condition, and hence it changes according to the departure time. Given a road r ∈ E, from junction u to junction v, and a departure time td, the average travel time on r at td is the average time it takes for vehicles to get from u to v, on r, when the departure time is td. The travel-time function, denoted fTT, is a function that maps each road r ∈ E, and a departure time td to the average travel time on r at td.1 This function can be used to find the fastest route between any two junctions in G, for any given departure time, e.g., using a variation of Dijkstra’s Algorithm. Several papers present methods for calculating the time-dependent fastest route between two given locations in a road network [8, 11, 15, 41]. Such methods can be applied either offline (for predefined departure times) or online. Similar methods are also used in industrial applications, e.g., the Bing Maps API2 allows calculating the fastest route between two locations based on live traffic data. Different companies, such as Google, Waze, TomTom, and others provide similar services.

Search network

Given a TARS query \(T=(\bar s, \bar t, \mathcal{Q}, O)\) over a dataset D, the answer sets of a temporal search subquery Qi, denoted Ai, is the set of objects of D that are relevant to Qi. We call the set \(P=\cup^m_{i=1}A_i\cup{s,t}\), of all the objects that are relevant to some search subquery, including the start and destination locations s and t, the points-of-interest (POIs, for short) of T and D.

A search networkSN is constructed from a TARS query and a road network by computing for the POIs P the travel-time function. Given two POIs, oi and oj, and a departure time td, the travel-time function fTT(oi,oj, td) returns the time it takes to drive from oi to oj, on the road network, when leaving oi at time td. In Section 5.1.3 we explain how an approximation of the travel-time function can be computed in real time and in a scalable fashion.

The inverted travel-time function \(f^i_{TT}\) returns for a triplet (oi,oj, ta) of an object oi, an object oj, and arrival time ta, the latest departure time from oi for which it is possible to get to oj at ta, when considering the traffic.

A search network SN = (P,fTT, d, e, l) includes the functions d, e and l that represent the expected stay duration, and the earliest and latest possible arrival times at each POI in P, respectively. The earliest and latest arrival times of a POI are a combination of the opening hours of the POI and the time constraints in the corresponding subquery. For example, if o represents a shoe store that is open between 10:00 to 17:00 and the user wishes to visit it not earlier than 12:00 and not later than 18:00, with an intention to spend 60 min there, we set the earliest and latest arrival times to be 12:00 and 16:00, respectively. For a subquery Qi = (qi,ei,li,di) and an object o ∈ Ai, we define e(o) and l(o) as follows. Let to(o) and tc(o) denote the opening and closing hours of o. Then, e(o) = max(to(o), ei) and l(o) = min(tc(o) − di, li). For an object o, we refer to [e(o), l(o)] as its arrival time interval. If l(o) < e(o) then this interval is considered empty and the time constraint is unsatisfiable.

For convenience, we consider P as an array of N + 2 POIs, in which, the first index (0) is reserved for s and the last index (N + 1) is reserved for t. With a slight abuse of notation, oi, di, ei and li denote an index of a POI in P, its expected stay duration and its earliest and latest arrival times, respectively. This notation is used henceforth. The notations are summarized in Table 2.
Table 2

Notation table for a search network

Notations

Description

Ai

A set of D objects which represent an answer to Qi

\(P=\cup^m_{i=1}A_i\cup{s,t}\)

Represents the POIs that are relevant to a TARS query \(\mathcal{Q}\)

fTT(oi,oj, td)

A function that returns the time it takes to drive from oi to oj, on the road network, when leaving oi at time td

SN = (P,fTT, d, e, l)

The search network, where d, e and l represent functions of the the expected stay duration, and the earliest and latest possible arrival times at each POI in P, respectively

Route

A route over a search network SN is a sequence ρ = s, o1,..., on, t, where s,o1,...,on,t are POIs of SN. The arrival function, denoted α, maps each object of ρ to the time of the arrival at that object. Similarly, the departure function, denoted β, maps each object of ρ to the time of departure from that object. For the start and destination locations, the arrival time is equal to the departure time, that is α(s) = β(s) and α(t) = β(t). With a slight abuse of notation, we also refer to the triplet (ρ, α, β) as a route.

Given a TARS query \(T=(\bar s, \bar t, \mathcal{Q}, O)\), the restriction of ρ to Qi is the set \({\rho}_{|_{Q_i}} = A_i\cap\rho\), of the objects of ρ that are in the answer set of Qi.

To answer a TARS query, we need to satisfy all the subqueries and all the constraints of T. A route ρ = s, o1,..., on, tsatisfies a TARS query \(T=(s,t, \mathcal{Q}, O)\) if the following conditions hold.
  1. 1.

    Subqueries are satisfied: For each \((Q_i,\tau_i)\in \mathcal{Q}\), the probability that at least one object of the answer set of Qi will satisfy Qi is at least τi. That is, given that for each oj, its success probability is pj, \(\tau_i\leq 1-\prod_{o_j\in {\rho}_{|_{Q_i}}}(1-p_j)\).

     
  2. 2.

    Order constraints are satisfied: For every order constraint (Qi, Qj) ∈ O and every pair of objects \(o_{k_1}\in {\rho}_{|_{Q_i}}\) and \(o_{k_2}\in {\rho}_{|_{Q_j}}\), \(o_{k_1}\) appears in ρ before \(o_{k_2}\), i.e.,k1 < k2.

     
  3. 3.

    Temporal constraints are satisfied: (1) The arrival time at oi should be within its arrival time interval, that is, ei ≤ α(oi) ≤ li for every 1 ≤ i ≤ n. (2) The departure time for oi ≠ s is β(oi) = α(oi) + di and e0 ≤ β(s) ≤ l0. (3) The arrival at the target t should be within its arrival time interval, that is, et ≤ α(t) ≤ lt.

     
  4. 4.

    The travel time should comply with the traffic conditions: The time it takes to reach oi + 1 from object oi, at departure time td, should not be smaller than the actual travel time between these objects, at td. That is, for every oi in ρ, α(oi + 1) − β(oi) ≥ fTT(oi, oi + 1, β(oi)).

     
The overall travel time of a route ρ = s, o1,..., on, t is α(t) − β(s). The relevant notations are summarized in Table 3.
Table 3

Route notation table

Notations

Description

(ρ, α, β)

A route

ρ = s, o1,..., on, t

A sequence of objects over the search network SN

α

An arrival function which maps each object of ρ to its corresponding time of arrival

β

A departure function which maps each object of ρ to its corresponding time of departure

Answer to a TARS query

Given a TARS query T over a dataset D and a network N, an answer to T is a route ρ that satisfies T. An optimal answer to a query T is a route ρ that satisfies the following: (1) it is an answer to T, and (2) for every route ρ′ that is an answer to T, the overall travel time of ρ does not exceed the overall travel time of ρ′.

Estimating the stay duration

Estimating the stay duration is intricate. For some types of POIs, the duration of staying in entities that satisfy the user is different from the duration of staying in entities that do not satisfy the user. For example, compare the duration of dining at a restaurant to the delay at a restaurant that the user only checked out before deciding to have lunch in another place. For other types of POIs, the stay duration is constant. For example, a shopper is likely to spend similar time at a shoe store where he finds what he needs and at a shoe store that does not satisfy him.

We refer to these two types as satisfaction-dependent and satisfaction-independent stay durations. For types of entities with satisfaction-independent stay durations, we assign the given stay duration to each POI of that type. Determining whether a subquery is satisfaction-dependent can be done by examining historical data. Such data indicate the average stay durations at relevant objects and allow us to categorize subqueries into satisfaction-dependent and satisfaction-independent.

Modeling types of entities with satisfaction-dependent stay durations is difficult, because for such types the full stay duration will only be spent at a POI that satisfies the user, however, we do not know which one among the POIs of the type will be the satisfying one. To deal with this, we assume that the full stay duration is spent only at the first visited POI of the type and a short stay duration at the others.

Note that by assigning the full stay durations to the first POI of each type, we guarantee that if the user is satisfied with these POIs, she will reach the other POIs of the route no later than their latest allowed arrival-time interval (assuming that the actual stay duration in POIs is not greater than the estimated stay duration). However, assigning the full stay duration to any POI other than the first POI of each type would result in a misleading plan, where a user may reach a POI too late, i.e., later than the allowed arrival-time. To see this, consider an example of a route via restaurants and a museum. Suppose we assign a stay duration of an hour to the first restaurant on the route and indeed the user stays an hour in that restaurant before reaching the museum. Then, a stay of an hour prior to visiting the museum is already assumed in the calculations of the route. However, if the first restaurant is assigned only a five-minute stay duration, then an actual stay of an hour at this restaurant, may cause an hour-late arrival at the museum.

In this solution there are POIs of the same type with different stay duration. This can be achieved by rewriting the given TARS query such that subqueries which refer to satisfaction-dependent types are duplicated and an order constraint is added to specify that the entity of the first subquery should be visited before the entities of the second subquery. The first subquery is assigned a full stay duration while the second will have a short stay duration. Note that for each subquery, at most one duplication is needed. For example, consider a TARS query according to Example 1 that includes a search subquery Q4 for “vegetarian restaurant”. The rewriting of the query is done by adding a search subquery \(Q^{\prime}_4\) that also comprises the keywords “vegetarian restaurant”. The probability threshold of Q4 is set to be very low so that there will only be a single entity of \(Q^{\prime}_4\) in the produced route. An order constraint \((Q^{\prime}_4,Q_4)\) is added so that the restaurant of \(Q^{\prime}_4\) will be the first restaurant on the produced route. The stay duration for the restaurant of \(Q^{\prime}_4\) is set to be the duration of a lunch, say one hour, whereas the stay duration for the restaurants of Q4 is set to be the time it takes to examine a restaurant, say ten minutes. Figure 2 illustrates an example of such a query rewriting operation. Note that by duplicating a subquery, the same entity is expected to appear twice in the route. For instance, the same vegetarian restaurant will appear first due to the subquery \(Q^{\prime}_4\) and immediately after that as an answer to the subquery Q4 where the travel time from the restaurant to itself is zero. Such duplicate appearance is needed for a correct computation of the success probability of Q4 and it can simply be ignored when the route is presented to the user. All this is being done by the system. Thus, in the next sections we assume that it does not affect the formulation of queries and their evaluation.
Figure 2

Illustration of a query rewrite when a satisfaction-dependent subquery is detected

4 Algorithms

TARS is a generalization of the Traveling Salesperson Problem (TSP). That is, we can define any given TSP problem as a TARS as follows. Given a road network and a start location, as the input to a TSP problem, we build a TARS query in which each node of the given network is the unique answer, with probability one, to a different subquery, without any temporal or order constraints and with a travel-time function that is constant and is proportional to the distances between nodes. It is easy to see that a solution to the TARS query is also a solution to the TSP problem.

TARS also generalizes several variations of TSP, by considering travel times as constants. In the Generalized Traveling Salesperson Problem (GTSP) [38], the objects of a TSP problem are partitioned into categories and the goal is to find the shortest route while visiting a single object from each category. Thus, TARS where all the success probabilities are equal to one, having no temporal constraints and having no order constraints, is similar to GTSP. In a Prize Collecting TSP (PCTSP) [2], prize values are attached to the objects of a TSP problem and the goal is to find the shortest route for which the sum of the prizes on visited objects exceeds a given quota. Since prizes are similar to probabilities, a TARS query with a single subquery can express PCTSP. Providing a route that travels via a set of predefined addresses with temporal constraints, (assuming constant travel speeds) is similar to TSP with Time Windows [45]. Finally, TSP with Pickup and Delivery [21] deals with satisfying order constraints. In all these problems the goal is to find the shortest route and they are all NP-hard. TARS combines all these problems for the case of varying traffic conditions under the goal of finding an optimal departure time and the fastest route.

Since TARS generalizes NP-hard problems, we do not expect to find a polynomial-time algorithm for TARS, and we settle for polynomial-time heuristics. In this section we describe three heuristics to answer TARS queries. Throughout this section we assume that the TARS query has the form \(T=(\bar s, \bar t, \mathcal{Q}, O)\) and that it is posed over a data set D and a road network G, as presented in Section 3.

4.1 Greedy search (GS)

The first algorithm we present, as Algorithm 1, is a greedy-search algorithm. The algorithm consists of two nested loops—an outer loop over departure times and an inner loop of greedy extension steps. The outer loop iterates over possible departure times. Given eo and l0—the earliest and latest departure times from s—and a time interval δ, the algorithm generates the sequence \(\sigma = e_0, e_0+\delta, e_0+2\delta, \ldots, e_0+(\lfloor (l_0-e_0)/\delta\rfloor)\delta\) of departure times. It examines departure from s at any time in σ (Line 4 of Algorithm 1). For example, if e0 and l0 are 10:00 and 10:30, respectively, and δ is 10 min, then the algorithm iterates over the departure times 10:00, 10:10, 10:20, 10:30. (Note that we later examine the significance of δ and the manner by which its value is determined.) The answer is the shortest route among the candidate routes that are computed in the inner loop for the possible departure times. If there are no candidate routes the algorithm reports a failure by returning an empty route.

In the inner loop (beginning at Line 7 of Algorithm 1), the algorithm starts with a route s,t, comprising merely the source s and the target t. In each iteration, the algorithm extends the partial route that was built in previous iterations by calling the method ExtendPath (Line 8). It does so by adding a POI that satisfies those query constraints that are still unsatisfied, while it strives to minimize the overall travel time. This inner loop terminates when no further POIs can (or need to) be added to the route. In such case, R′ = ∅ in Line 9. At this stage, if the route satisfies the given TARS query, it becomes a candidate route. If its travel time is smaller than the travel times of routes computed in previous iterations, we keep it (by assigning it to R). Otherwise, it is discarded. Eventually, R is the route whose travel time is the smallest among the routes computed for different departure times.

Adding a POI to a partial route may cause other POIs in that route to become redundant. Consider, for example, a search query Qi that looks for a “restaurant” where the probability threshold is 0.8. Suppose that a POI o represents a restaurant with a success probability of 0.75, and o is added to the partial route at some iteration. Also consider that in a later iteration, the algorithm adds to the route another POI o′ that represents a restaurant and has a success probability of 0.85. (Note that in such scenario, o′ was not added first because adding it would have increased the overall travel time more than the addition of o). However, adding o′ causes o to become redundant, thus o can be removed. Therefore, after each extension step (see the sub-method ExtendPath), the algorithm checks if any of the existing objects can be removed without violating any constraint that has already been satisfied.

The sub-method ExtendPath receives a sequence ρ and extends it by adding to it an object. It does so by iterating over the relevant objects (Line 4) and all the positions in the route where the object can be added (Line 5). Only objects that contribute to the satisfaction of the query and do not violate the constraints are considered (Line 7). The addition for which the travel time of the constructed sequence R′ is the smallest is chosen and returned.

The value of δ affects the accuracy of the result. How large is this effect? In general, when computing a route using time interval δ′ (where δ′ < δ), the algorithm is expected to compute a route that is faster by at most δ − δ′ from the route computed using δ. For example, choosing δ to be 3 min instead of 10 min is expected to decrease travel time by no more than 7 min. Hence, the value of δ can be chosen based on the level of accuracy required by the user, i.e., setting the value of δ to be 10 min will likely be accurate enough for most practical purposes. The following lemmas present this formally.

Lemma 1 shows that for a specific route, a delay of δ in the departure time can decrease the overall travel time by no more than δ.

Lemma 1

Given a route ρ, let T1 be the fastest travel time on ρ when the departure time is t and let T2 be the fastest travel time on ρ when departing at time t + δ. Suppose that T2 < T1, then T1 − T2 ≤ δ.

Proof

The proof is by contradiction. Consider a route ρ = (o1, ..., on − 1, on). Suppose the fastest travel time of a user u1 that departs from o1 at t is longer by more than δ than the fastest travel time of a user u2 that departs from o1 at t + δ. This means that u2 arrives at on before u1. We examine two cases.

The first case (see Fig. 3) is when u1 and u2 travel on the same path from o1 to on, i.e., on exactly the same road segments of the network. In this case, there is a point p′ where u1 and u2 are at the same distance from on. We refer to this as the “meeting point”.
Fig. 3

Illustration of the first case in Lemma 1. User u1 begins the route at time t and user u2 begins the route at time t + δ. P′ denotes their meeting point

When two users arrive at different times at a road segment, and both travel at the maximal possible speed according to traffic, the one who arrives later cannot complete the traversal of the road segment before the first one (although it may spend less time traversing the segment).

From the meeting point, the travel time of u2 to on is smaller than the travel time of u1 to on. Since both users travel on the same path, u1 could travel from this point to on at the same travel speed as u2, so both users should arrive at the same time. This is a contradiction to the assumption that u2 arrives at on before u1. It is also a contradiction to T1 being the travel time of u1 (because in such case, u1 could have traveled faster).

The second case (see Fig. 4) is when u1 and u2 travel on different paths, via the same objects o1,..., on. In such a case, we define \(T^{\prime}_1\) to be the travel time on the path of u2 when departing at time t. The travel time T1 of u1 is not greater than \(T^{\prime}_1\) because T1 is the minimal travel time from o1 to on when departing at t. In addition, \(T^{\prime}_1-T_2\leq\delta\), as explained for the first case. Hence, \(T_1\leq T^{\prime}_1\leq T_2+\delta\), in contradiction to the assumption that T1 > T2 + δ (i.e., the assumption that u1 arrives after u2).□
Fig. 4

Illustration of the second case in Lemma 1

Lemma 2, which is concluded from Lemma 1, asserts that for a given TARS query, the optimal answer for departure at time t + δ can be faster by at most δ than the optimal answer for departure at t.

Lemma 2

Consider a TARS query Q. Suppose that ρ1 and ρ2 are the optimal answers to Q (fastest routes), for departure times t and t + δ, respectively. Let T1 and T2 be the travel times of ρ1 and ρ2. Then, T1 − T2 ≤ δ.

Proof

Let \(T^{\prime}_2\) be the travel time on route ρ2 with departure time t. Then, according to Lemma 1, \(T^{\prime}_2-T_2\leq\delta\). Since ρ1 is the fastest answer to Q for departure time t, it holds that \(T_1\leq T^{\prime}_2\). Hence, \(T_1 - T_2\leq T^{\prime}_2 - T_2\leq \delta\).□

Lemma 2 considers the effect of the departure time on the travel duration of optimal answers to TARS queries. However, the GS algorithm is merely a heuristics, and hence, it may not compute the optimal answer. The following proposition illustrates the effect of the departure time on the travel duration of routes computed by GS.

Proposition 1

Given a TARS query Q, suppose that \(\rho^{\textit{\tiny GS}}_1\) and \(\rho^{\textit{\tiny GS}}_2\) are the routes GS computes for the departure times t and t + δ, respectively. Let \(\rho^{\textit{\tiny opt}}_1\) and \(\rho^{\textit{\tiny opt}}_2\) be the optimal answers to Q, for departure times t and t + δ, respectively. Consider \(T^{\textit{\tiny GS}}_1\), \(T^{\textit{\tiny GS}}_2\), \(T^{\textit{\tiny opt}}_1\) and \(T^{\textit{\tiny opt}}_2\) to be the fastest travel times on routes \(\rho^{\textit{\tiny GS}}_1\), \(\rho^{\textit{\tiny GS}}_2\), \(\rho^{\textit{\tiny opt}}_1\) and \(\rho^{\textit{\tiny opt}}_2\). Then, \(T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny GS}}_2 \leq \delta + T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny opt}}_1\).

Proof

Since \(\rho^{\textit{\tiny opt}}_2\) is the optimal answer to Q, for departure time t + δ, \(T^{\textit{\tiny GS}}_2\geq T^{\textit{\tiny opt}}_2\). Hence, \(T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny GS}}_2 \leq T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny opt}}_2\). From Lemma 2, follows \(T^{\textit{\tiny opt}}_1 - T^{\textit{\tiny opt}}_2 \leq \delta\). Thus, \(T^\textit{\tiny GS}_1 - T^{\textit{\tiny opt}}_2 = T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny opt}}_1 + T^{\textit{\tiny opt}}_1 - T^{\textit{\tiny opt}}_2\leq T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny opt}}_1 + \delta\).□

The subexpression \(T^{\textit{\tiny GS}}_1 - T^{\textit{\tiny opt}}_1\) in Proposition 1 is the difference between the optimal travel duration and the travel on the route computed by GS. This difference is due to GS being a heuristics rather than an exact algorithm. Its size depends on the quality of the heuristics, not on δ. Thus, when increasing δ, the reduce in the accuracy of GS is at the size of the increase (i.e., at the size of the change in δ). Decreasing δ improves the accuracy, similarly.

4.2 One-pinned greedy search

GS works in a greedy fashion, and hence, it has no mechanism to escape a local minimum which can either lead to a failure in finding a solution or cause finding a sub optimal solution. To mitigate this problem we developed the, more exhaustive, 1-Pinned Greedy-Search Algorithm (1-PGS). Intuitively, 1-PGS forces GS to consider each POI as part of the answer. It does so by calling GS with initial partial routes of the form s,o,t, instead of applying GS with an initial route s,t. We refer to the object o as the pinned object. Iteratively, 1-PGS examines all the possible POIs as the pinned object, and it returns the route with the minimal overall travel time among all the routes it generated for the pinned objects.

In practice, we do not consider all the objects of the dataset as potential pinned objects. We only examine as pinned objects those objects that are located in areas relevant to the search. We elaborate on this in Section 5 when we explain how we implemented the algorithms.

4.3 Mixed integer linear programming

The 1-PGS algorithm is an improvement of GS. As such, it is guaranteed to always match or outperform the former. It can, in some cases, escape the local minimum problem that hinders GS, however, it may fail in doing so. Therefore, 1-PGS may not be able to find a solution when a solution exists or it may find a suboptimal solution. To ward off this problem it is necessary to use an approach that is based on a global search rather than a local greedy approach. A naive solution is using a brute-force exhaustive search. This is done by examining all the permutations of POIs for any feasible selection of POIs that can potentially satisfy the given TARS query and examining the possible departure times for each such route. Even without addressing the departure times, the number of options that need to be examined is large. At worst, in a map containing n POIs, all with a non-zero probability, the number of permutations such an algorithm would need to examine is n!. Even for very small values of n, say 15, this requires 15! > 1012 permutations. Examining a permutation requires checking if it represents a route that satisfies all the query constraints. Optimistically, assuming this process requires merely 100 nanoseconds, the entire computation would still require more than 4 years. As the number of POIs increases, even slightly, the exhaustive search quickly becomes far beyond the computational capabilities of any existing machine.

Since TARS is an intricate combinatorial optimization problems, formulating TARS requires a model that is highly expressive. Achieving a practical solution also requires the model to be accompanied with a powerful framework that can solve the evaluation problem within a reasonable time frame. Hence, we opt to model the TARS problem as a Mixed Integer Linear Program (MILP) whose solution yields an approximation of the optimal route. MILP is highly expressive and it can be computed using one of the many solvers that were developed for it.

There are a few difficulties with this approach we need to overcome. Firstly, MILP problems are NP-Hard, however, as a heuristic, we can limit the computation time and let the solver return the best solution it discovered within the limited time frame. Secondly, if the travel-time function is non-linear it is impossible to formulate it within the context of a linear program. To deal with this problem we use a few heuristics which will be explained in the following sections. Thirdly, approaches for modeling TSP problems as integer programming problems generally examine the entire map, and include a variable for each road in the network (see [36]). However, the number of roads in a city can be large (tens of thousands of roads). Thus, running a MILP solver with a variable for each road is unfeasible. We observe that the number of POIs in a search network is much smaller than the total number of nodes or roads in the entire road network. Hence, by using the search network defined in Section 3 and ignoring junctions and other non-POIs, we reduce the number of variables and constraints significantly. A TARS query typically contains about three or four search subqueries, and for each query we can choose the top ten objects with the highest probability in the area of the search. Modeling TARS this way allows reducing the number of objects in the problem to be around forty, or less.

To model the TARS problem as a MILP problem, we need to represent all the constraints of the problem as linear constraints. A MILP problem consists of: (1) a set of decision variables, some of which are limited to assignments of integer values only; (2) a set of linear constraints on the values of the decision variables; (3) a linear objective function that is being minimized (or maximized). The first step is linearizing the travel-time function fTT.

4.3.1 Linearizing the travel-time function

The travel-time function fTT returns an estimation of the time it takes to reach a given target POI from a given source POI when starting the travel at a specified time. For any two POIs oi and oj, the function fTT(oi,oj,td) returns, for a departure time td, the estimated travel time from oi to oj. This function is typically not linear. Since the constraints and the objective of a MILP expression must be specified using linear functions, we need to obtain, for every two POIs oi and oj, an approximated linear function for fTT(oi,oj,td). To achieve a good approximation, we restrict the function to a time interval \([e^{\prime}_i, l^{\prime}_i]\) that represents the possible times for traveling from oi to oj, according to the specifications of the problem.

Using sampling we select a set of points within the departure-time interval \([e^{\prime}_i, l^{\prime}_i]\) and produce a linear approximation of fTT(oi, oj, td) with respect to this interval. The following section explains how to compute the interval \([e^{\prime}_i, l^{\prime}_i]\).

4.3.2 Intervals of arrival and departure

To achieve a good approximation of fTT(oi,oj,td), we apply linear regression over time intervals that are as short as possible. This is done by considering, for each oi and oj, only the relevant times to travel from oi to oj, according to possible departure times, from oi. For example, if oi represents a restaurant that is open from 10:00 and the expected stay duration is one hour, we limit the departure time from oi to be not before 11:00. Similarly, we take into account restrictions of the user (e.g., the user wants to eat after 12:00), the order constraints and the minimal travel time to get to oi from the start location s. We also need to consider the departure time from oj and limit the interval so that arrival to oj will not be too late, according to the constraints of the problem.

Using merely the arrival time interval, which consists of opening times and TARS constraints, to compute these time intervals, provides a crude estimation. We can improve the estimation by considering for each POI its position in possible routes. To do so, if o is the k-th object in a route ρ then we say that o is in positionk in ρ. We denote by \(I_j^{(k)}\) the interval that represents the possible arrival time at oj in routes that contain oj in position k. Note that for some j and k, there is no possible route where POI oj is in position k. In such case, \(I_j^{(k)}\) is an empty set.

To compute the values of the time intervals, we apply a process where each interval induces constraints on other intervals, iteratively, as described next.

Let N be the number of nodes in the search network, other than s and t. Assuming that o0 is s and oN + 1 is t, the indexes of objects in the network are 0,..., N + 1. Let K be an upper bound on the number of objects in possible routes. (We discuss the estimation of K later.) We start by constructing the Arrival-Time Matrix (ATM), as presented in Fig. 5. The figure represents the initial possible arrival-time intervals of each POI oj, for 0 ≤ j ≤ N + 1, in each possible position 1 ≤ k ≤ K.
Fig. 5

Initial arrival-time matrix (ATM)

Every element \(I_j^{(k)}\) of the arrival-time matrix that is not denoted by an empty set has the form \([e_j^{(k)}, l_j^{(k)}]\) where \(e_j^{(k)}\) and \(l_j^{(k)}\) represent the earliest and latest arrival times at oj, respectively, in routes that contain oj in the k-th position. Initially, for each object oj, \(e_j^{(k)}=e_j\) and \(l_j^{(k)}=l_j\). Note that the first object of the route must be s (i.e., o0), and hence, \(I_j^{(1)}=\emptyset\) for j ≥ 1. Since s is only visited as the first POI, \(I_0^{(k)}=\emptyset\) for k ≥ 2. For m search subqueries, the route must include at least m + 2 POIs (m objects to satisfy the subqueries, s and t). Thus, the target t (i.e., oN + 1) can only be visited in position that is greater than m + 1. Hence, \(I_{N+1}^{(k)}=\emptyset\) for 1 ≤ k ≤ m + 1. Obviously, the last object of the route must be t, so \(I_j^{(K)}=\emptyset\) for j < N + 1.

We explain now how to compute the arrival-time intervals. Suppose that the arrival-time intervals for the k-th position are \(\{I_0^{(k)}, I_1^{(k)}, ..., I_{N+1}^{(k)}\}\), where \(I_j^{(k)}=[e_j^{(k)}, l_j^{(k)}]\), for 1 ≤ j ≤ N + 1. So, if a user arrives at a POI oj as the k-th object of the route, the earliest arrival time at oj is \(e_j^{(k)}\) and the earliest arrival time at oi as the k + 1-st POI is \(e_j^{(k)}+d_j+f_{TT}\left(o_j, o_i, e_j^{(k)}+d_j\right)\)—the earliest arrival time at oj plus the stay duration dj at oj plus the travel time from oj to oi, according to the travel-time function fTT, when leaving oj as early as possible, i.e., at \(e_j^{(k)}+d_{j}\). Consequently, the earliest arrival time at any oi as in the k + 1-st position is \(\min_{0\leq j\leq N+1}\left(e_j^{(k)}+d_j+f_{TT}\left(o_j, o_i, e_j^{(k)}+d_j\right)\right)\). This is because fTT is a monotonically increasing function (i.e., an increase in the departure time cannot cause a decrease in the arrival time.)

Algorithm 2 presents the iterative computation by which the intervals of the initial ATM are reduced. We apply a similar algorithm BackwardReducedATM to reduce the latest arrival times lj,k. The three main differences in BackwardReducedATM in comparison to Algorithm 2 are (1) the external iteration is from K − 1 down to 1, (2) we use the inverted travel-time function \(f^i_{TT}\), and (3) we replace max by min and replace min by max.

If after applying the two reduction algorithms, interval \(I_j^{(k)}\) is empty, then it is impossible to visit POI oj at position k. Furthermore, if for all the POIs of an answer set of a search query, all their time intervals are empty, then the TARS query is unsatisfiable. The corresponding departure-time interval for oj is \(I_j^{\prime(k)}=[e_j^{\prime(k)},l_j^{\prime(k)}]=[e_j^{(k)}+d_j, l_j^{(k)}+d_j]\).

Finally, for every pair of POIs oj, oi and 1 ≤ k ≤ K, we use linear regression to approximate the travel time function fTT(oi, oj, td) within the departure time interval \(I_j^{\prime(k)}\) (since \(t_d\in I_j^{\prime(k)}\)) and produce \(a^{(k)}_{i, j}t_d + b^{(k)}_{i, j}\).

4.3.3 Modeling TARS as a MILP problem

We continue now with our main goal of modeling TARS as a MILP problem. Table 4 defines the notations we use in the formulation. Note that we formulate the problem over a search network, which is typically small, and not over the entire dataset, as described in Section 3. To simplify the notation, we denote by \(\overline{0,n}\) the set 0,1,...,n.
Table 4

Notations for modeling a TARS query T over a search network \(\mathcal{N}\)

\(i\in\overline{0, N+1}\)

POI indexes, where 0 and N + 1 represent the start and target nodes

A = {A1, A2, ..., Am}

Answer sets of the subqueries of T

\(A_i=\{i_1, i_2, ..., i_{|A_i|}\}\)

Indexes of the POIs in the answer set Ai

\(O=\{(q, q')\mid q, q'\in \overline{1,m}\}\)

Order constraints specified using subquery indexes

pi, j ∈ (0, 1)

The probability of oj satisfying subquery Qi

\(a_{i, j}^{(k)}\beta(i)+b_{i, j}^{(k)}\)

Linearized earliest-arrival-time function at POI j as the route’s k-th POI where β(i) is the departure time from i

dj

Expected stay duration time at node j.

\(e_j^{(k)}, l_j^{(k)}\)

Earliest and latest allowed arrival times at node j as the route’s k-th POI

τi ∈ (0, 1)

Minimum probability threshold of subquery Qi

Defining the decision variables (I)

In Table 5 we present the straightforward approach to define the decision variables of the MILP problem.
Table 5

Decision variables (straightforward, inefficient)

\(\forall i, j\in\overline{0, N+1}:\)xi, j ∈ 0, 1

Variable xi,j = 1 iff the route goes from POI i to POI j

\(\forall i\in\overline{1, N+1}: \alpha_{i}\in[0, 1]\)

Arrival time at node i

β0 ∈ [0, 1]

Departure time from node 0, i.e., from s

A decision variable is defined for every pair of relevant POIs indicating whether the route traverses from the first POI to the second POI

The objective is to minimize the total travel time of the route. That is, to minimize αN + 1 − β0.

Before we present the constraints, let us examine the number of decision variables according to the definitions in Table 5. The number of variables is (N + 2)2 + (N + 1) + 1. Formulating the MILP problem using O(N2) decision variables is likely to significantly degrade the performance of any solver.

Defining the decision variables (II)

To reduce the number of decision variables and improve the efficiency, we present a different approach that is based on the use of an estimated upper bound K on the number of objects in the computed route. That is, we find K such that computed routes contain at most K POIs.

Table 6 presents the decision variables in this approach. The number of variables is 2·(N + 1)·K + 1. Note that it is affected by K, and hence, it is important to estimate K as accurately as possible—when K is too large, the computation is not efficient. When K is too small, we may find a suboptimal solution or may not find a solution at all.
Table 6

Decision variables (second attempt)

\(x_0^{(1)} = 1\)

A constant value indicating that a route should begin at the source POI

\(\forall i\in\overline{1, N+1}, \forall k\in\overline{1, K}:\)\(x_{i}^{(k)}\in{0, 1}\)

\(x_{i}^{(k)}=1\) iff POI i is in position k of the route

\(\forall i\in\overline{1, N+1}, \forall k\in\overline{1, K}:\)\(\alpha_{i}^{(k)}\in[0, 1]\)

Arrival time at POI i when it is in position k

\(\alpha_{0}^{(1)}=e_0\)

Source arrival time is equal to its earliest departure time

\(\beta_{0}^{(1)}\in[0, 1]\)

Departure time from the source

A decision variable is defined for each POI and each position of that POI in the constructed route

To compute K, we use a heuristic that estimates the expected number of POIs in constructed routes. Let \(p^{\prime}_i\) denote the harmonic mean of the probabilities of all the POIs in the answer set of the subquery Qi. The expression \((1-p^{\prime}_i)^{K_i}\) is the probability of failing Ki times to satisfy the user by objects whose probability is \(p^{\prime}_i\). Hence, \(1-(1-p^{\prime}_i)^{K_i}\) estimates the probability to satisfy subquery Qi by visiting Ki POIs. Thus, we require this probability to be at least τi—the probability threshold of Qi—that is \(1-(1-p^{\prime}_i)^{K_i}=\tau_i\). From this follows \(K_i=\log_{1-p^{\prime}_i}{(1-\tau_i)}\), that is \(K_i=\frac{\log(1-\tau_i)}{\log(1-p^{\prime}_i)}\). Finally, we define \(K=2+\sum_{i=1}^{m}K_i=2+\sum_{i=1}^{m}\frac{\log{(1-\tau_i)}}{\log{(1-p^{\prime}_i)}}\), where 2 is added to take into account the start and target locations.

Objective

Minimize the total travel time, that is,
$$ \textrm{minimize} \left(\sum\limits_{k=2}^K \alpha_{N+1}^{(k)}-\beta_0^{(1)}\right) $$

Next, we need to define the constraints of the MILP. The constraints are defined with the following aim. Any assignment of values, to the decision variables, that satisfies the constraints corresponds to a route that satisfies the given TARS query.

Linear constraints

  1. (1)
    The target t is the last node in the route.
    $$ \forall\; k\in\overline{1, K-1},\;\; x_{N+1}^{(k)}+\sum\limits_{i=1}^{N+1} x_i^{(k+1)}\leq 1 $$
    Note that for a given k, \(x_{N+1}^{(k)}=1\)ifft is in position k. In such case, the constraint requires \(x_{i}^{(k+1)}=0\) for every i, i.e., no object in position k + 1.
     
  2. (2)
    There is at most one node in each position.
    $$ \forall\; k\in\overline{1, K},\;\; \sum\limits_{i=1}^{N+1} x_i^{(k)}\leq 1 $$
     
  3. (3)
    A node cannot appear in more than one position.
    $$ \forall\; i\in\overline{0, N+1},\;\; \sum\limits_{k=1}^{K} x_i^{(k)}\leq 1 $$
    The target must be visited, thus, \(\sum\limits_{k=1}^{K} x_{N+1}^{(k)}= 1\).
     
  4. (4)
    There are no “empty” positions in the middle of the route, i.e., the number of nodes in position k is equal to the number of nodes in position k + 1, unless t is in k.
    $$ \forall\; k\in\overline{1, K},\;\; \sum\limits_{i=0}^N x_i^{(k)}=\sum\limits_{j=1}^{N+1}x_j^{(k+1)} $$
     
  5. (5)
    For each subquery, the probability of success is not below the threshold τi.
    $$ \forall\; i\in\overline{1, m},\;\; 1-\prod\limits_{j\in A_i}(1-p_{i, j})^{\sum\limits_{k=1}^K{x_j^{(k)}}}\geq\tau_i $$
    Note that (1 − pi, j) is the probability that j does not satisfy Qi, \(\sum_{k=1}^K{x_j^{(k)}}\) is 1 if j appears in the route and 0 otherwise. Thus, the product is the probability that all the visited objects of Ai failed, and by reducing it from 1, we receive the probability that at least one object satisfied Qi. This constraint is not linear, so we change it to be linear, as follows:
    1. (a)

      \( \prod_{j\in A_i}(1-p_{i, j})^{\sum_{k=1}^K{x_j^{(k)}}}\leq 1- \tau_i \)

       
    2. (b)

      \( \log\left(\prod_{j\in A_i}(1-p_{i, j})^{\sum_{k=1}^K{x_j^{(k)}}}\right)\leq \log\left(1-\tau_i\right)\)

       
    3. (c)

      \(\sum_{j\in A_i}\log\left((1-p_{i, j})^{\sum_{k=1}^K{x_j^{(k)}}}\right)\leq \log\left(1-\tau_i\right) \)

       
    4. (d)

      \(\sum_{j\in A_i}\sum_{k=1}^K{x_{j}^{(k)}}\log(1-p_{i, j})\leq \log\left(1-\tau_i\right)\)

       
     
  6. (6)
    Arrival times of visited POIs must be valid. \(\forall\; i\in\overline{1, N+1}, \forall\; k\in\overline{1, K}, \textit{if } I_i^{(k)}\neq\emptyset\)
    $$ x_i^{(k)}\cdot e_i^{(k)}\leq \alpha_i^{(k)}\leq x_i^{(k)}\cdot l_i^{(k)} $$
    Departure from the source s must be valid.
    $$ e_0^{(1)}\leq \beta_0^{(1)}\leq l_0^{(1)} $$
     
  7. (7)
    Arrival times must be consistent with the linearized arrival-time function. \(\forall\; j \in \overline{1, N+1} , \forall\; k\in\overline{2, K-1},\;\; \)
    $$ \alpha_j^{(k+1)} \geq \sum\limits_{i=1}^{N+1}\left(a_{i, j}^{(k)}\cdot(\alpha_i^{(k)}+d_i)+b_{i, j}^{(k)}\right)\cdot x_i^{(k)}\cdot x_j^{(k+1)} $$
    The sum on the right side of the equation is equal to
    $$ \sum\limits_{i=1}^{N+1}\left(a_{i, j}^{(k)}\cdot\alpha_i^{(k)}\cdot x_i^{(k)}+a_{i, j}^{(k)}\cdot d_i\cdot x_i^{(k)}+b_{i, j}^{(k)}\cdot x_i^{(k)}\right) x_j^{(k+1)}. $$
    From Constraints (6) follows \(\alpha_i^{(k)}=0\Leftrightarrow x_i^{(k)}=0\). Hence, \(x_i^{(k)}\cdot\alpha_i^{(k)}=\alpha_i^{(k)}\), and the above sum can be written as
    $$ \left(\sum\limits_{i=1}^{N+1}a_{i, j}^{(k)}\cdot\alpha_i^{(k)}+\sum\limits_{i=1}^{N+1}(a_{i, j}^{(k)}\cdot d_i+b_{i, j}^{(k)}) x_i^{(k)}\right) x_j^{(k+1)} $$
    The multiplication by \(x_j^{(k+1)}\) makes this equation non-linear. To solve this, note that when \(x_j^{(k+1)}=1\), the constraint should be
    $$ \alpha_j^{(k+1)} \geq \sum\limits_{i=1}^{N+1}a_{i, j}^{(k)}\cdot\alpha_i^{(k)}+\sum\limits_{i=1}^{N+1}(a_{i, j}^{(k)}\cdot d_i+b_{i, j}^{(k)}) x_i^{(k)} $$
    and when \(x_j^{(k+1)}=0\), the node j is not in position k + 1, so we do not need any constraint, i.e.,\(\alpha_j^{(k+1)} \geq 0\). We observe that \(\sum_{i=1}^{N+1}a_{i, j}^{(k)}\cdot\alpha_i^{(k)}\leq 1\) and \(\sum_{i=1}^{N+1}(a_{i, j}^{(k)}\cdot d_i+b_{i, j}^{(k)})\cdot x_i^{(k)}\leq 1\). Thus, to express the above two cases, the right side of the constraints can be formulated as
    $$ \sum\limits_{i=1}^{N+1}a_{i, j}^{(k)}\cdot\alpha_i^{(k)}+\sum\limits_{i=1}^{N+1}(a_{i, j}^{(k)}\cdot d_i+b_{i, j}^{(k)})\cdot x_i^{(k)}-2\cdot(1-x_j^{(k+1)}) $$
    Finally, we can write the constraint as a linear equation:
    $$ \alpha_j^{(k+1)}\geq\sum\limits_{i=1}^{N+1}a_{i, j}^{(k)}\cdot\alpha_i^{(k)}+\sum\limits_{i=1}^{N+1}(a_{i, j}^{(k)}\cdot d_i+b_{i, j}^{(k)})\cdot x_i^{(k)}+2\cdot x_j^{(k+1)}-2 $$
    Similarly, for \(j\in\overline{1, N+1}\) and k = 1,
    $$ \alpha_j^{(2)}\geq a_{0, j}\cdot\beta_0^{(1)}+b_{i, j}^{(k)}\cdot x_i^{(k)}+2\cdot x_j^{(k+1)}-2 $$
     
  8. (8)
    Order constraints must be satisfied \(\forall\;(q, q^{\prime})\in O, \forall\; k\in\overline{1, K},\;\;\)
    $$ {\textrm if} \sum\nolimits_{i\in A_{q^{\prime}}}x_i^{(k)}=1 {\textrm then} 1-\prod\nolimits_{k^{\prime}=1}^{k}\prod\nolimits_{j\in A_{q}}(1-p_{q, j})^{x_j^{(k^{\prime})}} \geq\tau_i. $$
    We linearize the constraints as we did for Constraints (5), and receive \(\sum_{j\in A_{q}}\sum_{k^{\prime}=1}^{k} x_j^{(k^{\prime})}\cdot \log(1-p_{q, j})-\log(1-\tau_i)\leq 0\). Now, we can remove the conditional part of the equation above by using a constant M = − 2·log(1 − τi), and we rewrite the equation in the following linear form:
    $$ \sum\limits_{j\in A_{q}}\sum\limits_{k^{\prime}=1}^{k} x_j^{(k')}\cdot \log(1-p_{q, j})-\log(1-\tau_i)\leq M\cdot\left(1-\sum\limits_{i\in A_{q^{\prime}}}x_i^{(k)}\right) \\ $$
    When \(\sum_{i\in A_{q^{\prime}}}x_i^{(k)}=1\), the left side of the equation is equal to 0 and this is the constraint we need. When \(\sum_{i\in A_{q^{\prime}}}x_i^{(k)}=0\), the condition of the “if” statement does not hold, so there is no need for a constraint. The constant M was chosen so that in such case, the inequality is always satisfied.
     

4.4 Complexity analysis

Let N denote the number of POIs in the TARS problem. Let \(d=\left\lfloor\frac{l_0-e_0}{\delta}\right\rfloor\) denote the total number of departure times examined by the algorithms. The time complexity of GS is O(N·d·K2), where K is the number of POIs in the constructed route. This is because the algorithm checks, for each possible POI, at most K insertion positions and it does so for d different departure times. In practice, however, d can be bounded by a constant, so the time complexity is O(N·K2). Algorithm 1-PGS, calls GS N times. Hence, its time complexity is O(N2·K2). For the MILP algorithm there are two stages to consider. The first is the formalization of the MILP problem, which comprises the following three steps. (1) Producing the time of departure intervals, by creating and reducing the ATM. This requires \(|\textrm{ATM}|=(N+2)\cdot K\) iterations. (2) Computing a linear approximating of the travel-time function, for every pair of distinct objects and for every 1 ≤ k ≤ K. This has O((N + 2)·(N + 1)·K·TC) time complexity, where TC is the time required for the linear regression process. Since TC is independent of N and K, we can consider it as a constant, so this step has O(N2 ·K) time complexity. (3) Constructing the 7 sets of linear constraints for the MILP solver requires producing O((N + 1)·K) constraints (see Constraint (5) and Constraint (6)). Hence, the overall time complexity of this stage is O(K·N2), and it is quadratic in N. The second stage to consider is solving the MILP problem. Solving a MILP problem has exponential time complexity in the number of variables. (Recall that in our model, we use 2·(N + 1)·K + 1 variables.) However, effective MILP solvers use advanced heuristics to compute a solution. Based on that, in Section 5 we show that by limiting the running time of the solver we can achieve good results within a reasonable time frame.

5 Experimental evaluation

In this section, we present an experimental evaluation of the algorithms that were presented in Section 4. We describe our experimental setting—the data and the methodology we used—and we analyze the results. Our goals are to compare the algorithms according to (1) their rate of success in finding a solution to given TARS queries, (2) their effectiveness, that is, the overall travel time of the computed routes, and (3) their efficiency, that is, the running time that it takes to compute a solution.

5.1 Setting

In our experiments we used the dataset and the queries that are presented below.

5.1.1 Dataset

We used the Yahoo Local Search API3 to generate the dataset. We posed, using this API, the following 7 search queries: (1) “ikea”, (2) “gas station”, (3) “pharmacy”, (4) “bank”, (5) “shoe store”, (6) “cinema” and (7) “post office”, limited to an area in the city of San Francisco, and retrieved the first 10 objects of each result. We denote these queries by Q1,..., Q7. Retrieving 10 objects from each result was based on the tendency of geographic search engines to provide results in batches of size 10 (e.g., see maps.google.com). There are additional, more sophisticated, methods for deciding which objects should serve as candidate POIs. For instance, previous papers have shown how to reduce the number of objects that need to be considered when answering a route-search query, including the use of spatial indexes [6, 32, 39]. Their methods can be combined with our algorithms for the step of constructing the search network. Furthermore, the user can also manually filter some of the search results which she considers as irrelevant to the search.

We assigned success probabilities to the objects based on their position in the search results, that is to say, if an object o1 precedes an object o2 in the search result then o1 was assigned a higher probability than o2. The reason for setting the probabilities in this way is that search engines rank the objects by their relevance to the search terms. That is, we strive to make the probabilities proportional to the relevance scores. The assigned probabilities were constructed in the range [0.4, 0.9] using the distribution function \(e^{-\gamma\cdot(i-1)}-(1-p_h)\), where \(\gamma=-\frac{ln(1+p_h-p_l)}{n_r-1}\), ph = 0.9, pl = 0.4, nr = 10, and 1 ≤ i ≤ 10 is the position of the object in the search result. This represents a behavior that is similar to the well known “long tail” phenomenon in search. The dataset that we used is available online as an XML document (see [31]).

5.1.2 Search queries

We generated a set of TARS queries, from the search queries Q1,..., Q7 as follows. First, we created a set of \({7 \choose 4}=35\) TARS queries of size 4 by constructing all the possible selections of 4 queries among Q1,..., Q7. The start and destination locations of each query were chosen arbitrarily in the area of San Fransisco. The time constrains for the start and destination locations, and the minimum probability threshold of each subquery are presented in Table 7. The durations where not set in the query. Instead, they were arbitrarily set for each POI to be within the range \((0,\textit{max-duration}]\). From this initial set of queries, we generated two groups of queries—queries with time constraints, denoted TC4, and queries without time constraints, denoted NTC4. The time constraints of the first group are provided in Table 7. Note that in the presence of time constraints, some queries do not have a solution. Accordingly, TC4 denotes the set of 25 queries, among the queries with time constraints, for which we were able to find a solution (using various methods). It is difficult to find a solution for queries of TC4, thus, we used this set to test the success rates of the algorithms and their effectiveness when handling queries with constraints that are not easy to satisfy. We used NTC4 to test the effectiveness of the algorithms on queries that are being satisfied relatively easily.
Table 7

Time constraints we used for the queries

Search query

Earliest arrival

Latest arrival

Max stay duration

Threshold (τ)

Source

10:00

12:00

0

Irrelevant

Target

10:00

18:30

0

Irrelevant

“Ikea”

11:30

14:30

90

0.95

“Gas station”

06:00

23:59

15

0.1

“Bank”

13:00

15:00

45

0.1

“Shoe store”

15:00

16:00

45

0.95

“Post office”

11:00

17:00

60

0.95

“Cinema”

16:00

16:05

120

0.1

“Pharmacy”

06:00

23:59

20

0.85

Earliest” and “Latest” refer to the arrival-time constraints. Stay max-duration refers to the maximal stay duration at POIs of this type and is given in minutes. Actual stay durations for POIs matching each of the above search terms where arbitrarily selected to be within \((0,\mbox{max-duration}]\)

Similarly, we created a set of \({7 \choose 5}=21\) TARS queries of size 5, i.e., queries that comprise 5 subqueries. We constructed from it two sets of queries—a set TC5 of queries with time constraints, and a set NTC5 of queries without time constraints. Among the 21 queries with time constraints, only to 11 queries we were able to find a solution. TC5 denotes the set of these 11 queries. Note that the queries of TC5 and NTC5 are larger and more complicated than the queries of TC4 and NTC4. The total number of queries we issued is \(|\emph{TC4}|+|\emph{NTC4}|+|\emph{TC5}|+|\emph{NTC5}|=25+35+11+21=92\).

5.1.3 Building a scalable travel-time function

To generate a travel-time function, we collected travel-time data for selected pairs of POIs, using the Bing Maps API.4 This API receives a start location and a destination. It returns the fastest route between these locations, at the time of the search, taking into account live traffic data. Collecting and storing the time it takes to travel from every possible location to every other possible location at any given departure time is not feasible. Instead, we implement a simple approximation method using a heuristic which is inspired by hierarchical shortest path algorithms (see [14, 22, 42]) and hierarchical networks [20].

Given a set of predefined POIs, we sampled the travel time between each pair, for different departure times. The measures were conducted in intervals of approximately k minutes for a period of 24 h. We refer to k as our sampling rate and it is a configurable parameter. Based on this sample, we created a travel-time function, that for any given hour and a pair of objects, returns the travel time between these objects at the given hour. We used linear interpolation to complete the travel-time function for departure times that were not measured.

For each pair of distinct POIs, the data contains a set of time-dependent travel-time samples in intervals of k minutes, for a 24-h time period. Hence, for a dataset of n objects, the number of time samples is 24·6·n·(n − 1) = 144(n2 − n). Therefore, using this approach for every possible pair of POIs in a city is not scalable. To provide scalability, we partitioned the city of San-Fransisco into 50 areas. In each area we arbitrarily selected 50 POIs and generated a travel-time function for every pair of POIs, as described above. Similarly, we choose the center of each area, and for each pair of centers, we constructed a travel-time function.

To obtain the travel-time for any given pair of POIs (o1, o2) and a departure time t based on the above partition, we conduct the following procedure. We begin by finding the two closest POIs to o1 and o2. Suppose that \(o^{\prime}_1\) and \(o^{\prime}_2\) are these points. Note that we use d to denote the network distance between two points. We next proceed as follows:
  1. 1.

    If \(o^{\prime}_1\) and \(o^{\prime}_2\) are in the same area, the approximated travel-time function for the pair (o1, o2) is defined as that of \((o^{\prime}_1, o^{\prime}_2)\) multiplied by the ratio of network distances between the points, i.e., by \(\frac{d\left(o_1, o_2\right)}{d\left(o^{\prime}_1, o^{\prime}_2\right)}\).

     
  2. 2.

    If \(o^{\prime}_1\) and \(o^{\prime}_2\) are not in the same area, let (c1, c2) be the centers of the areas in which they reside. The travel-time function is the sum of the travel-time functions for the pairs \((o^{\prime}_1, c_1)\), (c1, c2) and \((c_2, o^{\prime}_2)\) multiplied by the network distances between the points. That is, we multiply the travel time by the ratio \(\frac{d\left(o_1, o_2\right)}{d\left(o^{\prime}_1, c_1\right)+d\left(c_1, c_2\right)+d\left(c_2, o^{\prime}_2\right)}\).

     
Using this method we can approximate the travel time for any two POIs in a city, in a scalable fashion.

5.1.4 Environment

Our algorithms were implemented using the Microsoft .Net Framework. Our experiments were conducted on a computer with a 64 bit ICore 5 Dual Core Intel processor and with 4 GB of RAM. We used the Gurobi Optimizer5 version 4.01 for solving the Mixed Integer Linear Program that was presented in Section 4. This optimizer is a high-end library for math programming, capable of solving Mixed Integer Linear Programming problems.

5.2 Evaluating our travel-time function approximation

To test the effectiveness of our approximated travel-time function, as defined in Section 5.1.3, we ran the following experiments. We randomly choose 100 pairs of locations within the city of San-Francisco, and at arbitrary departure times, we issued queries against the Bing API to find the fastest route between them. We then approximated the arrival time by using our approximation algorithm, and compared the approximated travel time to the actual travel time obtained by the API. Note that our approximation algorithm used historical data which were one week old. We ran the same test for different sampling rates (i.e., for different values of k). We refer to the approximation algorithm, when run with a sampling rate of k, as Approx-k. In Table 8, we report the arithmetic mean, the standard deviation and the maximum error ratios. As a baseline for our algorithm, we also report the results of a basic approximation algorithm which simply scales the travel times according to the travel distances. To do so, it multiplies the distance by an optimal pre-calculated constant. The pre-calculated constant is the average speed in the test area, and it was 43.7 kilometers per hour, in our tests.
Table 8

Error ratios over 100 runs between the real and approximated travel times

 

Approx-2 (%)

Approx-5 (%)

Approx-10 (%)

Approx-30 (%)

Constant scaling (%)

Arithmetic mean

14.5

14.6

14.7

14.8

29.8

Std. deviation

10.2

10.3

10.3

10.5

17.7

Maximum

62.1

62.4

62.4

63.7

71.2

The results in Table 8 show that the approximation algorithm we use produces better approximations than the baseline constant scaling. They also show that the sampling rate has only a minor effect on the quality of the approximation.

When running the approximation algorithm, we need to load the relevant historical data into memory, only once. After loading the data, the average time it takes to calculate the approximated travel time, for any source, target and departure time, is around 37 ms.

5.3 Results

We applied the GS, 1-PGS and MILP algorithms on the data described above. For our travel time approximation algorithm we used a sampling rate of k = 10 min. Note that as our algorithms rely on the travel-time approximation algorithm, the ratios of the approximated arrival and departure times to the actual ones are similar to the results reported in Table 8. We tested the MILP algorithm with a time limit that was enforced on the Gurobi solver. This time limit was implemented using the callback mechanism of Gurobi, to monitor the amount of time that passed since the computation was initiated. We limited the running time of the solver to 5, 10, 30 and 480 s and named the algorithms MILP05, MILP10, MILP30 and MILP480, respectively. Figure 6 shows the success rate in finding a solution, for each of the algorithms. The figure refers to the query groups TC4 and TC5, which contain 25 and 11 solvable queries, respectively.
Fig. 6

For each algorithm, the average on the ratio of the success rate of the algorithm to the best computed success rate, for the queries of TC4 and TC5

Figure 7 presents the effectiveness of the algorithms. It shows the average ratio of the overall travel time of routes computed by MILP480 to those computed by the other algorithms. These results were obtained by applying the different algorithms on identical queries from the query groups NTC4 and NTC5 (i.e., queries without time constraints). These query groups contain 35 and 21 different queries each, respectively, and the results refer to the average ratio of the travel times. In this test, the queries can be satisfied easily. Thus, all the algorithms were successful in finding a solution to each one the 56 (35 + 21) queries.
Fig. 7

The average ratio of the travel times of the routes computed by each one of the algorithms to those of MILP480, for the queries of NTC4 and NTC5

Figures 6 and 7 show that MILP480 has the highest success rate and it is the most effective among the algorithms we tested. MILP30 is almost as effective as MILP480 in the cases we examined. Figure 7 also shows that, in this setting, there are diminishing improvements in the effectiveness when allocating larger time windows for MILP. Additional tests show that this trend continues even when allowing MILP to run for more than an hour. The main reason for this is that a near optimal solution is discovered after a relatively short period of time and the additional time is used for very mild improvements in the effectiveness of the route.

Further analysis of the results shows that MILP30 and MILP480 consistently dominate the other algorithms, both in terms of success rate and effectiveness. A comparison between 1-PGS, MILP05 and MILP10 shows that, on the average, MILP05 and MILP10 outperform 1-PGS in terms of effectiveness and success rate. In all cases, GS had the worst success rate and effectiveness among the tested algorithms.

To test the efficiency of the algorithms, we measured their running times. The results are depicted in Fig. 8. The figure shows the time it takes for each algorithm to compute the result route, from the moment the user issues her query. Note that, on one hand, the running-time limitations on MILP refer only to the Gurobi MILP solver. Hence, the time it takes to construct the problem for the solver, e.g., create the constraints, may cause the running time to be larger than the specified time limit. On the other hand, when MILP is limited to a certain number of seconds, it may require less time than the limit to complete the computation, because the optimal route can be discovered before reaching the time limit. The times presented in Fig. 8 are the averages over all the runs.
Fig. 8

The average running times of the algorithms when applied over the different query groups. Times are presented on a logarithmic scale

Some of the running times presented in in Fig. 8 may be considered too high for online systems, however, since TARS is being used for planning and an answer to a TARS query may be a route with a future departure time, route calculations do not always have to be instantaneous.

5.4 Additional tests

To verify that our results are general and not specific to one setting, we conducted additional tests. First, we computed our queries using different start locations and different destinations. The results we obtained in these tests were very similar to those we presented in the previous section. This shows that our results are not biased by the selection of specific start and destination locations. Secondly, we computed the queries with various order constraints. Adding order constraints decreased the running times of all the algorithms because it decreased the search space of possible solutions. Effectiveness and success rate were not affected by the addition of order constraints. We do not further elaborate on these experiments as they do not provide any additional insights.

We ran tests on datasets of different sizes, to examine how well each of the algorithms scales, in terms of running time, success rate and effectiveness. To that end, we used the datasets SF5, SF10 and SF20 that were produced by posing search queries over San Francisco, as described in Section 5.1.1, where in SF5 we only retrieved the top 5 results of each search, in SF20 we retrieved the top 20 results of each search, and SF10 is the dataset we used in the previous section. TA100 is a dataset that was constructed by posing search queries over a map of Tel-Aviv. It contains approximately 100 objects for each one of the 7 search queries.

Table 9 presents the performances of the algorithms on TA100. MILP05 and MILP10 are not presented in this table because their time limits did not enable them to achieve worthwhile results on datasets of this magnitude. The table shows that MILP480 significantly outperforms GS and 1-PGS, in terms of success rate and effectiveness. On TA100, the success rate of MILP30 is lower than that of 1-PGS, however, MILP30 is more effective than 1-PGS. This is because 1-PGS is forced to examine many objects as the pinned object. By that, it reaches solutions that include objects that MILP30 does not get to examine, due to its time limit. MILP30, on the other hand, examines many variations with those objects that are likely to appear in the route. So, in most cases, MILP30 will find a faster route and will not waste computation time on irrelevant objects. On some cases, 1-PGS will find a solution when MILP30 fails to do so because it explores those objects that MILP30 ignores.
Table 9

Performances of the algorithms on TA100

 

GS

1-PGS

MILP30

Success rate

0.2

0.75

0.66

Effectiveness

1.54

1.24

1.14

The first row presents the ratio of the success rates of the algorithms to the success rates of MILP480.

The second row presents the ratio of the travel times of the routes computed by the algorithms to the travel times of the routes computed by MILP480

On smaller datasets (SF5, SF10 and SF20), MILP30 and MILP480 always outperformed 1-PGS, because they were able to cover well the set of possible potential routes. MILP05 and MILP10 tend to outperform 1-PGS on SF5 and SF10. However, on SF20, 1-PGS outperforms MILP05 and MILP10, in terms of success rate and effectiveness. This is, again, because the time given to the solver in MILP05 and MILP10 was insufficient for examining enough routes to find a good solution. Note that in this case, the running time of 1-PGS on SF20 is much higher than the running times of MILP05 and MILP10. GS has the worst success rate and effectiveness in all cases, because it examines merely one option without a comprehensive view of the problem.

Figure 9 shows a comparison in terms of success rate and effectiveness of MILP05, MILP10 and MILP30 to MILP480 over the datasets SF5, SF10 and SF20. It shows that if sufficient time is given to the solver, an increase in the size of the dataset has only a mild effect on the success rate and on the effectiveness of the algorithm.
Fig. 9

The left figure shows the success rate of the different MILP versions. The right figure shows the ratio of the effectiveness of the tested algorithm to the effectiveness of MILP480

Figure 10 shows the running times of the algorithms when tested over datasets of different sizes. The results confirm that the running times of GS and 1-PGS tend to increase linearly and quadratically, in the number of POIs, respectively. Moreover, as the number of POIs increases, the time required for the MILP algorithm to formalize the MILP problem for the Gurobi solver increases as well. Hence, the running times of the restricted MILP versions change when the size of the dataset grows. In SF5, the time required to formalize the problem for MILP is approximately 450 ms, which is only 10 % of the time given to the solver in MILP05. The results show that the running time required for the MILP algorithm to formalize the MILP problem scales quadratically. This is due to the fact that most of this time is spent on linearizing the time-dependent arrival time function for every pair of objects. Yet, on TA100 (the larger dataset), MILP480 is roughly 2.5 times faster, on the average, than 1-PGS (11 min versus 28 min), and still consistently and significantly outperforms 1-PGS.
Fig. 10

Running times, in seconds, of the algorithms over datasets of different sizes. Times are presented on a logarithmic scale

5.5 Analysis of the results

Figure 6 shows that the success rate plummets when increasing the number of subqueries from 4 to 5. The reason to this is that the difficulty to satisfy the TARS queries increases when the number of subqueries increases, due to the increase in the number of constraints. Figure 6 also shows that the plummet of MILP05 and MILP10 is sharper than that of 1-PGS. This is because the size of the search network increases when the number of subqueries increases. Hence, a time limit of 5 or 10 s is insufficient to produce results of the same quality as in evaluation of smaller queries.

Figure 8 shows that an increase in query size (i.e., in the number of subqueries) increases the running time. This is not surprising, since, as mentioned earlier, larger queries result in larger search networks. Another observation is that the running times increase when queries do not have time constraints. On the one hand, these queries are easy to satisfy, which means that finding a route that satisfies the query constraints can be done more easily. On the other hand, choosing the most optimal route among the possible routes becomes more difficult because the number of candidate routes becomes larger.

We use the notation \(alg_1\preceq alg_2\) to indicate that alg2 outperforms alg1 in all cases, both in terms of success rate and effectiveness. In our experiments, we see that \(\textrm{GS}\preceq\textrm{1-PGS}\). This is obvious, because 1-PGS choses the best route from a set of routes that includes the route the GS returns. We also observe that \(\textrm{MILP05}\preceq\textrm{MILP10}\preceq\textrm{MILP30}\preceq\textrm{MILP480}\). Obviously, allowing MILP more processing time can only improve the results. The more interesting comparison is, therefore, between GS and 1-PGS to the different MILP versions. For SF5, SF10 and SF20 we have seen that \(\textrm{1-PGS}\preceq\textrm{MILP30}\). For TA100, this no longer holds, however, \(\textrm{1-PGS}\preceq\textrm{MILP480}\) does hold. A conclusion from all these cases and the running times in Fig. 10 is that for different settings we need to use different time limits for MILP in order to achieve good success rate, effectiveness and efficiency, in comparison to 1-PGS.

The GS algorithm is highly efficient and scalable—its running time is linear in the number of POIs. However, in most cases, it produces results that are much worse than the results of the other algorithms, in terms of success rate or effectiveness. Hence, GS is only useful on huge datasets in which it becomes unfeasible to run the other algorithms.

Another important conclusion from the fact that MILP480 outperforms the other algorithms is that the linear approximation of the arrival-time function, defined in Section 4.3.1, and the approximation of K (the upper bound for the number of POIs in the route), are accurate enough in practice.

5.6 Illustration of an actual search

The following example illustrates the complexity of traffic-aware route search and the difference between the algorithms, even on a dataset of only 5 objects.

Example 2

Consider the TARS query that is depicted in Fig. 11, over the area of San Francisco. The start location is the point indicated by the green balloon marked by S and the destination is at the point indicated by the green balloon marked by D. Suppose that the user needs to be at a specific bank branch (Citibank at Rhode Island) at 7:00. The position of the bank is indicated by the red (or blue) square with a dollar sign on it. In addition, the user should visit a coffee shop among the two alternatives for coffee shops that are indicated by the squares with an icon of a coffee cup. The expected stay duration in the bank is 90 min and for the coffee shop, the expected stay duration is 60 min.
Fig. 11

An illustration of different routes that were computed by GS and MILP on a dataset containing 5 objects. The dollar icon denotes a bank and the coffee icons denote coffee shops

In this scenario, GS provides a route that typical users may naively choose. The planed route is indicated by the blue symbols. It reaches the bank at 7:00 and the coffee shop at 9:20. This route requires an overall travel time of 3:29 h. The route that MILP computed, goes via the places that are marked by red symbols. It reaches the bank at 7:00, goes to the near coffee shop at 8:34 and arrives at the destination with an overall travel time of 2:55 h, that is, more than half an hour earlier than the first option.

Example 2 is based on real travel-time data and it shows how taking into account traffic data can significantly affect travel times even for daily routine travels. More importantly, it shows that merely using travel times, e.g., using a greedy approach, is insufficient, and thus, elaborate algorithms such as MILP are needed.

6 Related work

In this section, we compare our work to similar, theoretical and practical, studies in the area of route search. The most known related problem is the Traveling Salesperson Problem (TSP). It is the problem of finding a minimum cost Hamiltonian cycle on a given graph [16]. TSP has many variations, one of which is the Generalized Traveling Salesperson Problem (GTSP) [38]. This variation has some properties in common with the problem of finding an answer to a TARS query. In GTSP, the vertexes of the graph are partitioned into sets, and the goal is to find the shortest route that visits a single object of each set. Both TSP and GTSP are NP-Hard problems. Answering a TARS query is a more intricate problem than solving TSP or GTSP. First, the travel time on the edges changes during the travel. Secondly, GTSP does not include temporal constraints or order constraints. Thirdly, to satisfy a TARS query it is sometimes necessary to traverse via multiple objects of a set due to the uncertainty whether objects satisfy the search requirements. Furthermore, in terms of complexity, approaches for modeling TSP problems as Integer Programming Problems [29, 36] generally examine the entire map, including all the roads. In Section 3 we showed that such models cannot be used for solving TARS problems efficiently.

Several other TSP variations have also been studied [2, 12, 21, 45], however, none of them can be used to answer a TARS query. Moreover, papers that have dealt with TSP variations in the past did not provide methods for modeling actual search problems as TSP variants.

There are various approaches for computing static (i.e., time-independent) and dynamic (i.e., time-dependent) travel speeds on urban roads, in the presence of traffic [3, 4]. Several papers studied the problem of finding the fastest route between two given locations when travel times on roads vary [8, 11, 15, 41, 48], or considered the costs of turns in route-planning tasks [5, 28, 46]. Tian et al. [44] studied the problem of finding a minimal cost path from a source location to a destination in a road network with cost updates. They propose PathMon, an efficient system for monitoring minimal cost paths in dynamic road networks. Several papers study personalized routing where the system should learn individual driving preferences of the user [30], take into account different criteria when considering the cost of a path [35] or handle cases where the user needs to perform tasks during the travel from the origin to the destination [1]. Many papers dealt with the task of planning escape routes, for evacuation in case of a disaster [26, 27, 33, 34, 49, 50].

Answering a TARS query is much more difficult than finding the fastest route between two locations because there are different constraints to satisfy and because the problem is NP-hard. This is illustrated in Example 2 which also shows that being able to calculate the fastest route from one location to another does not guarantee that the overall route will also be optimal in terms of travel time. Scheduling problems over networks with varying travel speeds have also been investigated [47], however, their work does not deal with the need to visit specific types of entities when traveling from one location to another.

Recently there has been a growing interest in the subject of answering route queries. Some papers (e.g., [10, 17, 32, 39, 40, 43]), have dealt with non-probabilistic datasets, while others (e.g., [9, 23, 24, 25, 37]) have dealt with uncertainty pertaining to the user satisfaction with the objects being visited. However, past work, only dealt with the problem of finding the shortest route. In comparison, this paper presents an overall solution, by modeling and dealing with travel durations, in a way that provides a mechanism for (1) planning the fastest route (along with its departure time), and (2) handling temporal constraints.

7 Conclusion

In this paper we presented the Traffic Aware Route Search (TARS) problem. In TARS, a user provides a search query containing free form search terms, time constraints and order constraints, and the goal is to find the fastest route that satisfies all the constraints. TARS queries are useful for planning both simple and complex travels, while taking into account varying traffic conditions, temporal constraints (such as opening hours of institutions and services) and restrictions on the order by which geographical entities should be visited in the travel.

We presented three heuristic algorithms, two of which are based on a greedy approach, either locally (GS) or globally (1-PGS), and a more elaborate algorithm (MILP) that heuristically formulates the problem as a Mixed Integer Linear Program and uses a solver to compute a solution. We tested the algorithms using real traffic data obtained using the Bing Maps API and actual POIs obtained using the Yahoo! Local Search API. An analysis of the results shows that MILP is the superior algorithm among the three algorithms we presented, both in terms of success rate and in terms effectiveness, i.e., finding the fastest route. The analysis also shows that MILP maintains its superiority on datasets of various sizes, and it does so while being more efficient than 1-PGS over large datasets. The GS algorithm is very efficient and also scales well for larger datasets, however, it is significantly outperformed by the other algorithms when considering effectiveness and success rate.

Finally, since MILP is effective, especially when the time limit is high, it can also serve as a baseline for testing scalable heuristic algorithms for TARS. That is, one can run MILP with a large time limit, say a few hours, and compare its results to the results of the algorithms whose effectiveness is tested.

As future work, we plan to investigate TARS in interactive setting, in which the search is conducted using a mobile device and the user can provide feedbacks as she visits the POIs. We intend to examine algorithms that use dynamic programming for this task.

Footnotes

  1. 1.

    In practice, the travel-time function depends on the date, e.g., the travel-time function for workdays may not be the same as the one for weekends, however, this is a technical issue, which we ignore to simplify the model. It can be handled by using different time functions according to the day of the travel.

  2. 2.
  3. 3.
  4. 4.
  5. 5.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Taub Building, Technion Israel Institute of TechnologyHaifaIsrael

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