Planning Your Route: Where to Start?
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Abstract
Tour planning is an important part of locationbased applications. A tour planner provides an optimized path through places of interests (targets) by minimizing the tour length or by applying some other constraints. It is usually formulated as a travelling salesman problem (TSP) or vehicle routing problem (VRP). In the present study, we focus on how to choose the best starting location in case of an openloop TSP. We consider three different strategies for selecting the starting location and compare their effectiveness with regard to optimizing tour length. If all targets are visible, most humans tend to start on the convex hull or from the furthest point. However, there are also cases where not all targets are visible beforehand, and the only information given is the bounding box. An optimum tour then typically starts from the corner or the shorter side of the box. Humans also have a strong preference to start from a corner. A good strategy can result in the shortest tour, while a bad strategy can even add 20% to the total tour length.
Keywords
Orienteering Locationbased game Tour planning Tour optimization TSP Human performance OMopsiIntroduction
The increasing popularity of sharing personal contents including photos, videos, and locations via social media has triggered an increasing interest in multiple recommendation systems. This interest has spawned an area of research that plays a vital role in building smart tourism. A survey by Borràs et al. (2014) reveals different web and mobilebased tourism recommendation systems whereas Gavalas et al. (2014) focused mainly on comparing mobilebased recommendation systems. These studies consider two challenges: (1) how to collect the content from social media, i.e., the places of interest (POI) for tourism, and (2) how to plan tours between the collected POIs. In this work, we focus on the second challenge by studying human problemsolving skills when applied to tour planning.
Automatic recommendation systems aim at providing tours by considering context awareness, personalization, and suitability for a tourist trip. Yu and Chang (2009) and Lim et al. (2018) provided personalized tours based on the interest, need, and preference of the individual user. Majid et al. (2013) studied recommendation systems utilizing the Flickr image database. While the TripPlanner software (Chen et al. 2015) adds the users’ preferred venues iteratively to a candidate tour, eCOMPASS (Gavalas et al. 2015) favors public transit aiming to minimize the environmental impact. Both systems can be used for realtime tour planning. Keler and Mazimpaka (2016) provide safety routing by avoiding areas within a city that are considered dangerous.
Several studies have also focused on optimizing tours using explicit criteria. De Choudhury et al. (2010) recommended trips based on popularity within a restricted time budget. Gionis et al. (2014) provided tour recommendations based on fixed start and end points while accounting for a specific time or distance budget. Bolzoni et al. (2014) proposed clusteringbased tour planning. Mor and Dalyot (2018) studied how to calculate distanceoptimized walking tours using a bidimensional nearest neighbor (NN) algorithm based on geotagged photos from social media. Li et al. (2017) combined both travel time and ride comfort in their tour planner. All these approaches considered the travel planning as an orienteering problem (Vansteenwegen et al. 2011).
Classical orienteering explores the navigational skill of participants. Mopsi orienteering (OMopsi) is a mobile orienteering game (Fränti et al. 2017) where the targets are realworld objects such as POIs in a smart tourism system. Unlike classical orienteering, OMopsi does not have a predefined visiting order of the targets. Consequently, finding the optimal tour corresponds to solving an openloop travelling salesman problem, which has been shown to be a nondeterministic polynomialtime (NP) hard problem (Papadimitriou 1977). This means that largescale instances cannot be solved by a computer in a reasonable amount of time. However, smallscale instances can make a good puzzle for humans to solve. Other popular puzzle games that are computationally hard are Sudoku (ErcseyRavasz and Toroczkai 2012) and Minesweeper (Scott et al. 2011).
In this paper, we study different strategies for selecting the best starting point of an openloop travelling salesman problem and study how human players perform in this. We statistically evaluate the goodness of various starting locations. We present three strategies to make the choice and compare their performances in terms of gap, aspect ratio, and how frequently they result in the optimum tour.
Human Performance
Most research has studied human performance in solving the TSP with varying problem size and the number of points on the convex hull. Researchers have also focused on designing algorithms that best match human performance.
Linear Relationship
Human skills in solving the TSP have been widely studied. MacGregor and Chu (2011) reported that humans can outperform simple TSP algorithms for relatively small TSP instances. Graham et al. (2000) showed that the time needed by a human to solve a TSP is linearly proportional to the size of the problem and that the gap to the optimal solution grows very slowly with the number of targets. Dry et al. (2006) made similar observations and found that the average time needed by a human to solve TSPs was linearly or nearlinearly related to its size. Vickers et al. (2003a) showed that human performance worsened when more points were located on the convex hull.
Modeling Human Behavior
Several researchers have tried to model the human capacity for problemsolving. MacGregor et al. (1999) compared three heuristics and found that the convex hull heuristic was the best fit for human approaches to solving the TSP. They suggested that people solve problems using a globaltolocal perceptual process. According to this concept, they proposed an algorithm (MacGregor et al. 2000). Graham et al. (2000) found that none of the five algorithms they studied was an adequate model of the mental process involved in human TSP solving. Instead, they proposed a hierarchical algorithm, which is closer to the psychological process of the human problem solver. Pizlo et al. (2006) later refined this algorithm and showed that it produces solutions that came very close to those produced by humans.
Van Rooij et al. (2003) postulated the crossing avoidance hypothesis. They claimed that humans are intuitively aware that a tour with crossing trajectories is not optimal. Therefore, humans typically avoid crossing trajectories in their optimal tour planning for the TSP. The same authors also claimed that there was a lack of evidence to support the convex hull hypothesis of MacGregor et al. (1999, 2000). Nevertheless, MacGregor et al. (2004) restated that the convex hull hypothesis provides a stronger correlation with human performances than the crossing avoidance hypothesis.
In a detailed analysis of both the globaltolocal and localtoglobal approaches, Vickers et al. (2003a, 2003b) observed that humans typically prefer to solve a TSP through the localtoglobal approach such as the nearest neighbor technique. Their results are also applicable to the openloop scenario. However, they found no evidence that humans would prefer the convex hull approach in the openloop case. Graham et al. (2000) pointed out that the applicability of the convex hull approach was limited to the closedloop TSP and did not extend to the openloop problem.
Convex Hull
This heuristic constructs a convex hull of all unvisited points. At each step, the next point is chosen such that the path never crosses this convex hull. This process is repeated until all the points have been visited. Macgregor et al. (2006) concluded that the convex hull approach was closer to human performance than either of two other heuristics they examined (crossing avoidance and nearest neighbor). They only used rather simple heuristics yielding results that were inferior to results produced by humans. While path length captures an important aspect of the solution, it merely reflects the goodness of the algorithm and is not indicative of human behavior. Macgregor et al. (2006) also measured the similarity of the paths by counting how many arcs the solutions shared. It shows that the convex hull correlated better to human behavior than to the plain nearest neighbor heuristic. In addition, this heuristic is somewhat nonhuman and it would be surprising if humans were constructing convex hulls in their head while solving the problems. It would be more humanlike to apply the nearest neighbor heuristic with some level of additional intelligence to avoid crossings and “dead ends.” Wiener et al. (2009) claimed that human performance is better than a pure nearest neighbor strategy. Therefore, the algorithm that correlates best with human behavior is still unknown.
In the following, we will merely focus on how to select the start point. Studies in this are very sparse in literature. Furthermore, players also need to plan based on the bounding box instead of the convex hull or nearest neighbor.
Where to Start?

in the middle

in a corner

at the short side of the box
Game Area
If the aspect ratio (AR) (the ratio of game area width to height) is less than 1, the game area is rotated by 90° to ensure that the horizontal edge always corresponds to the long edge. AR is used as an additional indicator as the starting point of the optimum route is likely to lie on the short side if the game area is narrow (high AR values).
Solving the Optimum Tour
Probability of a terminal point being located in a given cell
Cells  Probability  

A priori  Observed  
Any corner  4  16%  46% 
Any short side  6  24%  30% 
Middle  9  36%  7% 
Effect of the Road Network
Computer Performance
Datasets used in this study
Dataset  Type  Distance  Instances  Sizes  

OMopsi^{a}  Open loop  Haversine  Total  Low AR < 0.8  Medium AR = 0.8–1.2  High AR > 1.2  4–27 
147  43  49  55 
Location of the Terminal Points
Probabilities of strategies for different game area aspect ratios (AR)
Low AR < 0.8 (%)  Medium AR = 0.8–1.2 (%)  High AR > 1.2 (%)  

Corner  48  46  45 
Short edge  31  21  37 
Long edge  17  23  16 
Middle  4  10  2 
Game expansion with varying widths of game areas
Low AR < 0.8 (%)  Medium AR = 0.8–1.2 (%)  High AR > 1.2 (%)  

Corner to opposite corner/edges  58  29  50 
Corner to same corner/adjacent edges  12  29  20 
Short edge to other edges  23  20  25 
Others  7  22  5 
Player’s Start Position
Here, we include an extra location to the targets of a game to simulate a player’s playing. We consider each 25 grid cells as a potential starting location for a player, and with respect to the tour length, we find the best one among them. Overall optimum tour tends to have terminal points at the corners. It is therefore expected that the best start position among the 25 grid cells is very likely to be there.
By calculating the probabilities of grid cells to be the worst starting position, we observe that corners are also the most likely (45%) the worst start location. Thus, it is possible that for a particular game layout one corner may be the best starting location while another corner may be the worst. If a player is unlucky, then he/she could choose the worst corner. Evidently, corners are risky locations to start. Apart from corners, also the long edge (32%) and middle (17%) are likely to contain the worst starting position, while the short edge is the least likely (6%) to contain the worst starting position.
In order to examine how bad the worst location is, we calculate the gap (%). On average, the worst starting locations resulted in a 16% longer tour than the best start with no significant dependence on AR (18% for low, 14% for medium, and 16% for high AR). However, medium AR games have the lowest gap value. Therefore, for square games, the difference between the worst and the best start is the least noticeable, which makes these games tougher to predict the best start.
Comparing different starting point selection strategies
Probability to find best (%)  Gap to best (%)  

Random  3  8 
Middle  9  9 
Any corner  32  7 
Any short edge  23  6 
Human Results
Test Setup
Experimental data was collected from volunteers (students) participating in the Design and Analysis of Algorithms course at the School of Computing at the University of Eastern Finland in September 2018 (http://cs.uef.fi/pages/franti/asa). Two types of game setup were designed: visible (targets are visible) and blind (only bounding box is visible). In the visible setup, the students were given 90 instances (one at a time) and instructed to select the point where they thought the optimal tour would start. From this starting location, Concorde solver then computed the optimal tour. We then calculated the gap between the optimum tour with the studentselected fixed starting location and the overall optimum tour for the given target set. In the blind setup, the students saw only the bounding box but not the individual target locations. In our analyses, we include results from students who participated in both tasks (visible and blind) and had completed more than 60 tasks overall.
Results of the human experiments both for the visible and for the blind tasks. Rounds means tasks completed. Solved means number of times optimal solution found. Gap means average gap of the tour length and the optimal tour. Hull means number of times started on the convex hull. Furthest means number of times started from the furthest point. Corner means number of times started from the corner
Top group  

Visible  Blind  
Rounds  Solved  Gap  Hull  Furthest  Rounds  Solved  Gap  Corner 
90  78  0.3%  100%  58%  86  31  3.6%  93% 
90  77  0.3%  99%  51%  90  35  2.9%  99% 
90  75  0.4%  97%  42%  90  40  3.0%  100% 
90  77  0.4%  98%  39%  90  37  3.1%  97% 
65  53  0.5%  100%  55%  89  35  3.2%  95% 
90  74  0.5%  99%  49%  90  33  3.5%  95% 
90  76  0.5%  98%  42%  90  39  2.9%  98% 
89  70  0.5%  93%  35%  90  27  4.0%  56% 
90  78  0.7%  93%  42%  90  42  2.7%  100% 
90  67  0.7%  98%  40%  90  38  2.9%  91% 
90  63  0.8%  90%  29%  90  30  3.4%  83% 
90  68  0.8%  99%  37%  90  43  2.9%  98% 
90  66  0.8%  98%  49%  90  23  3.8%  88% 
90  63  0.9%  97%  44%  90  41  2.8%  93% 
89  64  0.9%  100%  55%  90  36  3.4%  79% 
90  57  1.0%  99%  34%  90  37  3.3%  100% 
90  62  1.0%  92%  31%  90  40  2.6%  100% 
90  64  1.0%  91%  44%  90  37  3.3%  99% 
90  58  1.1%  99%  44%  90  37  3.1%  83% 
90  64  1.1%  91%  44%  90  41  3.0%  98% 
90  58  1.6%  96%  20%  90  28  4.1%  74% 
90  55  2.3%  90%  36%  90  32  2.9%  100% 
Average  0.8%  96%  42%  Average  3.2%  92%  
Bottom group  
Visible  Blind  
Rounds  Solved  Gap  Hull  Furthest  Rounds  Solved  Gap  Corner 
78  24  3.3%  59%  15%  90  34  3.5%  82% 
90  17  4.2%  37%  11%  90  43  3.0%  100% 
87  14  5.0%  46%  10%  90  38  2.7%  100% 
90  14  5.6%  29%  7%  90  42  2.9%  92% 
90  11  5.6%  29%  7%  90  37  3.4%  93% 
79  9  5.7%  32%  5%  90  6  6.8%  3% 
90  6  5.8%  24%  1%  90  44  2.5%  100% 
90  4  6.0%  18%  2%  90  39  2.5%  99% 
Average  5.2%  34%  7%  Average  3.4%  84% 
Visible Setup
Most students achieved an average gap of less than 1%. To further investigate human performance, we calculate two additional parameters for each game: (i) convex hull and (ii) the furthest point from the center. We label each point whether it is on the convex hull or not. MacGregor (2012) suggested that humans may tend to start from a point on the convex hull. Our results confirm this. Starting on the convex hull also strongly correlates with performance. The top group almost always selected starting points on the convex hull (96% of times), while the bottom group used this strategy less frequently (34%).
Students strongly preferred to choose the furthest point in the game layout as the starting point. The top group chose it for 42% of cases. Although only a few students could explicitly describe their selection strategy when asked to do so, most of them told that they chose “most obvious outlier” point as the starting location. One student described his strategy was choosing the “leftmost or rightmost point whichever was further.”
Blind Setup
All students who were in the top group for the visible setup performed significantly worse in the blind setup. Furthermore, there is almost no difference in performance between players and no correlation between the visible and blind performances. The average gaps are 3.2% (top group) and 3.4% (bottom group) which is only slightly better than if the start point was chosen randomly (4.0%). This indicates that the skills required for the blind setup have very little in common with the skills required for the visible setup. Either the skill sets for both variants are completely different, or the blind setup requires more time to learn the necessary skill.
Here, most players found the computerpreferred corner point strategy. Top group students applied the corner strategy in 92% of times. The results also show a clear correlation between performance and the corner point strategy. Those players (17) who selected a corner point > 95% times achieved an average gap of 2.9% whereas the rest (13) had an average gap of 3.7%. None of the players seemed to use the short edge or any other strategy. One player chose the middle point strategy in 91% of games which resulted in a clear outlier (6.8% gap).
There was no observable correlation between the start point strategy and the game area aspect ratio.
In conclusion, human solvers were generally unable to discover any sophisticated strategy with such short experience in the game. Our expert players (the three authors) used this short edge strategy slightly more often (35%) but still relied mostly on the corner point strategy (62%) despite knowing these strategies beforehand.
Size of Instances
Vickers et al. (2003a) showed that humans performed worse when the game had more points on the convex hull. We tested this hypothesis as well by dividing the games into two groups: one with a high number of points on the convex hull and one with a small number. We did not find any significant differences between both groups.
Correlation to Study Performance
Finally, we compare human performance against their grades in the Design and Analysis of Algorithms course. Students were divided into two groups (high and low) according to their exam score. We tested two alternative hypotheses. Our primary hypothesis is that those who performed well in the course would also perform well in the TSP problemsolving.
Pearson correlation ratios between different factors with the amount of games solved
Affecting factor  All  Top group 

Convex hull  0.97  0.38 
Furthest point  0.93  0.53 
Course performance  0.11  0.72 
Conclusions
We have studied different strategies for selecting a start point for solving the openloop travelling salesman problem. The results showed that games have almost equal chance of having the best starting point at any grid location, except corners, which were most often the best choice. At the same time, corners can also be the worst choice, which makes it also the riskiest choice. Most human players in our trial used solely the corner point strategy although the short edge strategy would have been a slightly better choice.
With visible targets, the choice of human players clearly correlated with the starting point being on the convex hull or being the furthest from the center although a few were able to formulate and particularly argument to justify their choices. When increasing the number of targets, the player performance started to slightly degrade when measured by how many times the optimal solution was found. However, using the gap as the sole measure of success can be misleading. In our case, it would falsely imply the games become easier with the increasing number of targets, but this is clearly not the case. The average gap merely shows that the performance difference becomes less significant when the problem size increases.
Footnotes
Notes
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.