Worst Case and Probabilistic Analysis of the 2Opt Algorithm for the TSP
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Abstract
2Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every \(p\in\mathbb{N}\), a family of L _{ p } instances on which 2Opt can take an exponential number of steps.
Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]^{2}, where it was shown that the expected number of steps is bounded by \(\tilde{O}(n^{10})\) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1]^{ d }, for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2Opt improvement path of \(\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})\). When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to \(\tilde{O}(n^{4+1/31/d}\cdot\phi^{8/3})\). If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by \(\tilde{O}(n^{41/d}\cdot\phi)\). In addition, we prove an upper bound of \(O(\sqrt[d]{\phi})\) on the expected approximation factor with respect to all L _{ p } metrics.
Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ ^{ d }.
Keywords
TSP 2Opt Probabilistic analysis1 Introduction
In the traveling salesperson problem (TSP), we are given a set of vertices and for each pair of distinct vertices a distance. The goal is to find a tour of minimum length that visits every vertex exactly once and returns to the initial vertex at the end. Despite many theoretical analyses and experimental evaluations of the TSP, there is still a considerable gap between the theoretical results and the experimental observations. One important special case is the Euclidean TSP in which the vertices are points in \(\mathbb {R}^{d}\), for some \(d\in\mathbb{N}\), and the distances are measured according to the Euclidean metric. This special case is known to be NPhard in the strong sense [15], but it admits a polynomial time approximation scheme (PTAS), shown independently in 1996 by Arora [1] and Mitchell [13]. These approximation schemes are based on dynamic programming. However, the most successful algorithms on practical instances rely on the principle of local search and very little is known about their complexity.
The 2Opt algorithm is probably the most basic local search heuristic for the TSP. 2Opt starts with an arbitrary initial tour and incrementally improves this tour by making successive improvements that exchange two of the edges in the tour with two other edges. More precisely, in each improving step the 2Opt algorithm selects two edges {u _{1},u _{2}} and {v _{1},v _{2}} from the tour such that u _{1},u _{2},v _{1},v _{2} are distinct and appear in this order in the tour, and it replaces these edges by the edges {u _{1},v _{1}} and {u _{2},v _{2}}, provided that this change decreases the length of the tour. The algorithm terminates in a local optimum in which no further improving step is possible. We use the term 2change to denote a local improvement made by 2Opt. This simple heuristic performs amazingly well on “reallife” Euclidean instances like, e.g., the ones in the wellknown TSPLIB [17]. Usually the 2Opt heuristic needs a clearly subquadratic number of improving steps until it reaches a local optimum and the computed solution is within a few percentage points of the global optimum [7].
There are numerous experimental studies on the performance of 2Opt. However, the theoretical knowledge about this heuristic is still very limited. Let us first discuss the number of local improvement steps made by 2Opt before it finds a locally optimal solution. When talking about the number of local improvements, it is convenient to consider the state graph. The vertices in this graph correspond to the possible tours and an arc from a vertex v to a vertex u is contained if u is obtained from v by performing an improving 2Opt step. On the positive side, van Leeuwen and Schoone consider a 2Opt variant for the Euclidean plane in which only steps are allowed that remove a crossing from the tour. Such steps can introduce new crossings, but van Leeuwen and Schoone [20] show that after O(n ^{3}) steps, 2Opt finds a tour without any crossing. On the negative side, Lueker [12] constructs TSP instances whose state graphs contain exponentially long paths. Hence, 2Opt can take an exponential number of steps before it finds a locally optimal solution. This result is generalized to kOpt, for arbitrary k≥2, by Chandra, Karloff, and Tovey [3]. These negative results, however, use arbitrary graphs that cannot be embedded into lowdimensional Euclidean space. Hence, they leave open the question as to whether it is possible to construct Euclidean TSP instances on which 2Opt can take an exponential number of steps, which has explicitly been asked by Chandra, Karloff, and Tovey. We resolve this question by constructing such instances in the Euclidean plane. In chip design applications, often TSP instances arise in which the distances are measured according to the Manhattan metric. Also for this metric and for every other L _{ p } metric, we construct instances with exponentially long paths in the 2Opt state graph.
Theorem 1
For every p∈{1,2,3,…}∪{∞} and \(n\in\mathbb {N}=\{1,2,3,\ldots\}\), there is a twodimensional TSP instance with 16n vertices in which the distances are measured according to the L _{ p } metric and whose state graph contains a path of length 2^{ n+4}−22.
For Euclidean instances in which n points are placed independently uniformly at random in the unit square, Kern [8] shows that the length of the longest path in the state graph is bounded by O(n ^{16}) with probability at least 1−c/n for some constant c. Chandra, Karloff, and Tovey [3] improve this result by bounding the expected length of the longest path in the state graph by O(n ^{10}logn). That is, independent of the initial tour and the choice of the local improvements, the expected number of 2changes is bounded by O(n ^{10}logn). For instances in which n points are placed uniformly at random in the unit square and the distances are measured according to the Manhattan metric, Chandra, Karloff, and Tovey show that the expected length of the longest path in the state graph is bounded by O(n ^{6}logn).
We consider a more general probabilistic input model and improve the previously known bounds. The probabilistic model underlying our analysis allows different vertices to be placed independently according to different continuous probability distributions in the unit hypercube [0,1]^{ d }, for some constant dimension d≥2. The distribution of a vertex v _{ i } is defined by a density function f _{ i }:[0,1]^{ d }→[0,ϕ] for some given ϕ≥1. Our upper bounds depend on the number n of vertices and the upper bound ϕ on the density. We denote instances created by this input model as ϕperturbed Euclidean or Manhattan instances, depending on the underlying metric. The parameter ϕ can be seen as a parameter specifying how close the analysis is to a worst case analysis since the larger ϕ is, the better can worst case instances be approximated by the distributions. For ϕ=1 and d=2, every point has a uniform distribution over the unit square, and hence the input model equals the uniform model analyzed before. Our results narrow the gap between the subquadratic number of improving steps observed in experiments [7] and the upper bounds from the probabilistic analysis. With slight modifications, this model also covers a smoothed analysis, in which first an adversary specifies the positions of the points and after that each position is slightly perturbed by adding a Gaussian random variable with small standard deviation σ. In this case, one has to set \(\phi=1/(\sqrt{2\pi}\sigma)^{d}\).
We prove the following theorem about the expected length of the longest path in the 2Opt state graph for the three probabilistic input models discussed above. It is assumed that the dimension d≥2 is an arbitrary constant.
Theorem 2
 (a)
is O(n ^{4}⋅ϕ) for ϕperturbed Manhattan instances with n points.
 (b)
is O(n ^{4+1/3}⋅log(nϕ)⋅ϕ ^{8/3}) for ϕperturbed Euclidean instances with n points.
Usually, 2Opt is initialized with a tour computed by some tour construction heuristic. One particular class is that of insertion heuristics, which insert the vertices one after another into the tour. We show that also from a theoretical point of view, using such an insertion heuristic yields a significant improvement for metric TSP instances because the initial tour 2Opt starts with is much shorter than the longest possible tour. In the following theorem, we summarize our results on the expected number of local improvements.
Theorem 3
 (a)
is O(n ^{4−1/d }⋅logn⋅ϕ) on ϕperturbed Manhattan instances with n points when 2Opt is initialized with a tour obtained by an arbitrary insertion heuristic.
 (b)
is O(n ^{4+1/3−1/d }⋅log^{2}(nϕ)⋅ϕ ^{8/3}) on ϕperturbed Euclidean instances with n points when 2Opt is initialized with a tour obtained by an arbitrary insertion heuristic.
In fact, our analysis shows not only that the expected number of local improvements is polynomially bounded but it also shows that the second moment and hence the variance is bounded polynomially for ϕperturbed Manhattan instances. For the Euclidean metric, we cannot bound the variance but the 3/2th moment polynomially.
In [5], we also consider a model in which an arbitrary graph G=(V,E) is given along with, for each edge e∈E, a probability distribution according to which the edge length d(e) is chosen independently of the other edge lengths. Again, we restrict the choice of distributions to distributions that can be represented by density functions f _{ e }:[0,1]→[0,ϕ] with maximal density at most ϕ for a given ϕ≥1. We denote inputs created by this input model as ϕperturbed graphs. Observe that in this input model only the distances are perturbed whereas the graph structure is not changed by the randomization. This can be useful if one wants to explicitly prohibit certain edges. However, if the graph G is not complete, one has to initialize 2Opt with a Hamiltonian cycle to start with. We prove that in this model the expected length of the longest path in the 2Opt state graph is O(E⋅n ^{1+o(1)}⋅ϕ). As the techniques for proving this result are different from the ones used in this article, we will present it in a separate journal article.
As in the case of running time, the good approximation ratios obtained by 2Opt on practical instances cannot be explained by a worstcase analysis. In fact, there are quite negative results on the worstcase behavior of 2Opt. For example, Chandra, Karloff, and Tovey [3] show that there are Euclidean instances in the plane for which 2Opt has local optima whose costs are \(\varOmega(\frac{\log n}{\log\log n})\) times larger than the optimal costs. However, the same authors also show that the expected approximation ratio of the worst local optimum for instances with n points drawn uniformly at random from the unit square is bounded from above by a constant. We generalize their result to our input model in which different points can have different distributions with bounded density ϕ and to all L _{ p } metrics.
Theorem 4
Let \(p\in\mathbb{N}\cup\{\infty\}\). For ϕperturbed L _{ p } instances, the expected approximation ratio of the worst tour that is locally optimal for 2Opt is \(O(\sqrt[d]{\phi})\).
The remainder of the paper is organized as follows. We start by stating some basic definitions and notation in Sect. 2. In Sect. 3, we present the lower bounds. In Sect. 4, we analyze the expected number of local improvements and prove Theorems 2 and 3. Finally, in Sects. 5 and 6, we prove Theorem 4 about the expected approximation factor and we discuss the relation between our analysis and a smoothed analysis.
2 Preliminaries
An instance of the TSP consists of a set V={v _{1},…,v _{ n }} of vertices (depending on the context, synonymously referred to as points) and a symmetric distance function \(\mathsf {d}\colon V\times V\to\mathbb{R}_{\ge0}\) that associates with each pair {v _{ i },v _{ j }} of distinct vertices a distance d(v _{ i },v _{ j })=d(v _{ j },v _{ i }). The goal is to find a Hamiltonian cycle of minimum length. We also use the term tour to denote a Hamiltonian cycle. We define \(\mathbb{N}=\{1,2,3,\ldots\}\), and for a natural number \(n\in\mathbb{N}\), we denote the set {1,…,n} by [n].
 (a)
d(x,y)=0 if and only if x=y (reflexivity),
 (b)
d(x,y)=d(y,x) (symmetry), and
 (c)
d(x,z)≤d(x,y)+d(y,z) (triangle inequality).
A wellknown class of metrics on \(\mathbb{R}^{d}\) is the class of L _{ p } metrics. For \(p\in\mathbb{N}\), the distance d _{ p }(x,y) of two points \(x\in\mathbb{R}^{d}\) and \(y\in\mathbb{R}^{d}\) with respect to the L _{ p } metric is given by \(\mathsf{d}_{p}(x,y) = \sqrt[p]{x_{1}y_{1}^{p}+\cdots+x_{d}y_{d}^{p}}\). The L _{1} metric is often called Manhattan metric, and the L _{2} metric is wellknown as Euclidean metric. For p→∞, the L _{ p } metric converges to the L _{∞} metric defined by the distance function d _{∞}(x,y)=max{x _{1}−y _{1},…,x _{ d }−y _{ d }}. A TSP instance (V,d) with \(V\subseteq\mathbb{R}^{d}\) in which d equals d _{ p } restricted to V is called an L _{ p } instance. We also use the terms Manhattan instance and Euclidean instance to denote L _{1} and L _{2} instances, respectively. Furthermore, if p is clear from context, we write d instead of d _{ p }.
A tour construction heuristic for the TSP incrementally constructs a tour and stops as soon as a valid tour is created. Usually, a tour constructed by such a heuristic is used as the initial solution 2Opt starts with. A wellknown class of tour construction heuristics for metric TSP instances are socalled insertion heuristics. These heuristics insert the vertices into the tour one after another, and every vertex is inserted between two consecutive vertices in the current tour where it fits best. To make this more precise, let T _{ i } denote a subtour on a subset S _{ i } of i vertices, and suppose v∉S _{ i } is the next vertex to be inserted. If (x,y) denotes an edge in T _{ i } that minimizes d(x,v)+d(v,y)−d(x,y), then the new tour T _{ i+1} is obtained from T _{ i } by deleting the edge (x,y) and adding the edges (x,v) and (v,y). Depending on the order in which the vertices are inserted into the tour, one distinguishes between several different insertion heuristics. Rosenkrantz et al. [18] show an upper bound of ⌈logn⌉+1 on the approximation factor of any insertion heuristic on metric TSP instances. Furthermore, they show that two variants which they call nearest insertion and cheapest insertion achieve an approximation ratio of 2 for metric TSP instances. The nearest insertion heuristic always inserts the vertex with the smallest distance to the current tour (i.e., the vertex v∉S _{ i } that minimizes \(\min_{x\in S_{i}}\mathsf {d}(x,v)\)), and the cheapest insertion heuristic always inserts the vertex whose insertion leads to the cheapest tour T _{ i+1}.
3 Exponential Lower Bounds
In this section, we answer Chandra, Karloff, and Tovey’s question [3] as to whether it is possible to construct TSP instances in the Euclidean plane on which 2Opt can take an exponential number of steps. We present, for every \(p\in\mathbb{N}\cup\{\infty\}\), a family of twodimensional L _{ p } instances with exponentially long sequences of improving 2changes. In Sect. 3.1, we present our construction for the Euclidean plane, and in Sect. 3.2 we extend this construction to general L _{ p } metrics.
3.1 Exponential Lower Bound for the Euclidean Plane
In Lueker’s construction [12] many of the 2changes remove two edges that are far apart in the current tour in the sense that many vertices are visited between them. Our construction differs significantly from the previous one as the 2changes in our construction affect the tour only locally. The instances we construct are composed of gadgets of constant size. Each of these gadgets has a zero state and a one state, and there exists a sequence of improving 2changes starting in the zero state and eventually leading to the one state. Let G _{0},…,G _{ n−1} denote these gadgets. If gadget G _{ i } with i>0 has reached state one, then it can be reset to its zero state by gadget G _{ i−1}. The crucial property of our construction is that whenever a gadget G _{ i−1} changes its state from zero to one, it resets gadget G _{ i } twice. Hence, if in the initial tour, gadget G _{0} is in its zero state and every other gadget is in state one, then for every i with 0≤i≤n−1, gadget G _{ i } performs 2^{ i } state changes from zero to one as, for i>0, gadget G _{ i } is reset 2^{ i } times.
Every gadget is composed of 2 subgadgets, which we refer to as blocks. Each of these blocks consists of 4 vertices that are consecutively visited in the tour. For i∈{0,…,n−1} and j∈[2], let \(\mathcal{B}^{i}_{1}\) and \(\mathcal{B}^{i}_{2}\) denote the blocks of gadget G _{ i } and let \(A^{i}_{j}\), \(B^{i}_{j}\), \(C^{i}_{j}\), and \(D^{i}_{j}\) denote the four points \(\mathcal{B}^{i}_{j}\) consists of. If one ignores certain intermediate configurations that arise when one gadget resets another one, our construction ensures the following property: The points \(A^{i}_{j}\), \(B^{i}_{j}\), \(C^{i}_{j}\), and \(D^{i}_{j}\) are always visited consecutively in the tour either in the order \(A^{i}_{j} B^{i}_{j} C^{i}_{j} D^{i}_{j}\) or in the order \(A^{i}_{j} C^{i}_{j} B^{i}_{j} D^{i}_{j}\).
Due to the aforementioned properties, we can describe every nonintermediate tour that occurs during the sequence of 2changes completely by specifying for every block if it is in its short state or in its long state. In the following, we denote the state of a gadget G _{ i } by a pair (x _{1},x _{2}) with x _{ j }∈{S,L}, meaning that block \(\mathcal{B}^{i}_{j}\) is in its short state if and only if x _{ j }=S. Since every gadget consists of two blocks, there are four possible states for each gadget. However, only three of them appear in the sequence of 2changes, namely (L,L), (S,L), and (S,S). We call state (L,L) the zero state and state (S,S) the one state. In order to guarantee the existence of an exponentially long sequence of 2changes, the gadgets we construct possess the following property.
Property 5
If, for i∈{0,…,n−2}, gadget G _{ i } is in state (L,L) (or (S,L), respectively) and gadget G _{ i+1} is in state (S,S), then there exists a sequence of seven consecutive 2changes terminating with gadget G _{ i } being in state (S,L) (or state (S,S), respectively) and gadget G _{ i+1} in state (L,L). In this sequence only edges of and between the gadgets G _{ i } and G _{ i+1} are involved.
Lemma 6
If, for i∈{0,…,n−1}, gadget G _{ i } is in the zero state (L,L) and all gadgets G _{ j } with j>i are in the one state (S,S), then there exists a sequence of 2^{ n+3−i }−14 consecutive 2changes in which only edges of and between the gadgets G _{ j } with j≥i are involved and that terminates in a state in which all gadgets G _{ j } with j≥i are in the one state (S,S).
Proof
We prove the lemma by induction on i. If gadget G _{ n−1} is in state (L,L), then it can change its state with two 2changes to (S,S) without affecting the other gadgets. This is true because the two blocks of gadget G _{ n−1} can, one after another, change from their long state \(A^{n1}_{j}C^{n1}_{j}B^{n1}_{j}D^{n1}_{j}\) to their short state \(A^{n1}_{j}B^{n1}_{j}C^{n1}_{j}D^{n1}_{j}\) by a single 2change. Hence, the lemma is true for i=n−1 because 2^{ n+3−(n−1)}−14=2.
Now assume that the lemma is true for i+1 and consider a state in which gadget G _{ i } is in state (L,L) and all gadgets G _{ j } with j>i are in state (S,S). Due to Property 5, there exists a sequence of seven consecutive 2changes in which only edges of and between G _{ i } and G _{ i+1} are involved, terminating with G _{ i } being in state (S,L) and G _{ i+1} being in state (L,L). By the induction hypothesis there exists a sequence of (2^{ n+2−i }−14) 2changes after which all gadgets G _{ j } with j>i are in state (S,S). Then, due to Property 5, there exists a sequence of seven consecutive 2changes in which only G _{ i } changes its state from (S,L) to (S,S) while resetting gadget G _{ i+1} again from (S,S) to (L,L). Hence, we can apply the induction hypothesis again, yielding that after another (2^{ n+2−i }−14) 2changes all gadgets G _{ j } with j≥i are in state (S,S). This concludes the proof as the number of 2changes performed is 14+2(2^{ n+2−i }−14)=2^{ n+3−i }−14. □
In particular, this implies that, given Property 5, one can construct instances consisting of 2n gadgets, i.e., 16n points, whose state graphs contain paths of length 2^{2n+3}−14>2^{ n+4}−22, as desired in Theorem 1.
3.1.1 Detailed Description of the Sequence of Steps
Observe that the configurations 2 to 7 do not have the property mentioned at the beginning of this section that, for every block \(\mathcal{B}^{i}_{j}\), the points \(A^{i}_{j}\), \(B^{i}_{j}\), \(C^{i}_{j}\), and \(D^{i}_{j}\) are visited consecutively either in the order \(A^{i}_{j} B^{i}_{j} C^{i}_{j} D^{i}_{j}\) or in the order \(A^{i}_{j} C^{i}_{j} B^{i}_{j} D^{i}_{j}\). The configurations 2 to 7 are exactly the intermediate configurations that we mentioned at the beginning of this section.
If gadget G _{ i } is in state (L,L) instead of state (S,L), a sequence of steps that satisfies Property 5 can be constructed analogously. Additionally, one has to take into account that the three involved blocks \(\mathcal{B}^{i}_{1}\), \(\mathcal{B}^{i+1}_{1}\), and \(\mathcal{B}^{i+1}_{2}\) are not consecutive in the tour but that block \(\mathcal{B}^{i}_{2}\) lies between them. However, one can easily verify that this block is not affected by the sequence of 2changes, as after the seven 2changes have been performed, the block is in the same state and at the same position as before.
3.1.2 Embedding the Construction into the Euclidean Plane
The only missing step in the proof of Theorem 1 for the Euclidean plane is to find points such that all of the 2changes that we described in the previous section are improving. We specify the positions of the points of gadget G _{ n−1} and give a rule as to how the points of gadget G _{ i } can be derived when all points of gadget G _{ i+1} have already been placed. In our construction it happens that different points have exactly the same coordinates. This is only for ease of notation; if one wants to obtain a TSP instance in which distinct points have distinct coordinates, one can slightly move these points without affecting the property that all 2changes are improving.
 1.
Start with the coordinates of the points of gadget G _{ i+1}.
 2.
Rotate these points around the origin by 3π/2.
 3.
Scale each coordinate by a factor of 3.
 4.
Translate the points by the vector (−1.2,0.1).
From this construction it follows that each gadget is a scaled, rotated, and translated copy of gadget G _{ n−1}. If one has a set of points in the Euclidean plane that admits certain improving 2changes, then these 2changes are still improving if one scales, rotates, and translates all points in the same manner. Hence, it suffices to show that the sequences in which gadget G _{ n−2} resets gadget G _{ n−1} from (S,S) to (L,L) are improving because, for any i, the points of the gadgets G _{ i } and G _{ i+1} are a scaled, rotated, and translated copy of the points of the gadgets G _{ n−2} and G _{ n−1}.
3.2 Exponential Lower Bound for L _{ p } Metrics
We were not able to find a set of points in the plane such that all 2changes in Lemma 6 are improving with respect to the Manhattan metric. Therefore, we modify the construction of the gadgets and the sequence of 2changes. Our construction for the Manhattan metric is based on the construction for the Euclidean plane, but it does not possess the property that every gadget resets its neighboring gadget twice. This property is only true for half of the gadgets. To be more precise, we construct two different types of gadgets which we call reset gadgets and propagation gadgets. Reset gadgets perform the same sequence of 2changes as the gadgets that we constructed for the Euclidean plane. Propagation gadgets also have the same structure as the gadgets for the Euclidean plane, but when such a gadget changes its state from (L,L) to (S,S), it resets its neighboring gadget only once. Due to this relaxed requirement it is possible to find points in the Manhattan plane whose distances satisfy all necessary inequalities. Instead of n gadgets, our construction consists of 2n gadgets, namely n propagation gadgets \(G_{0}^{P},\ldots,G_{n1}^{P}\) and n reset gadgets \(G_{0}^{R},\ldots,G_{n1}^{R}\). The order in which these gadgets appear in the tour is \(G_{0}^{P}G_{0}^{R}G_{1}^{P}G_{1}^{R}\ldots G_{n1}^{P}G_{n1}^{R}\).
In the initial tour, only gadget \(G^{P}_{0}\) is in state (L,L) and every other gadget is in state (S,S). With similar arguments as for the Euclidean plane, we can show that gadget \(G_{i}^{R}\) is reset from its one state (S,S) to its zero state (L,L) 2^{ i } times and that the total number of steps is 2^{ n+4}−22.
3.2.1 Embedding the Construction into the Manhattan Plane
As in the construction in the Euclidean plane, the points in both blocks of a reset gadget \(G_{i}^{R}\) have the same coordinates. Also in this case one can slightly move all the points without affecting the inequalities if one wants distinct coordinates for distinct points. Again, we choose points for the gadgets \(G_{n1}^{P}\) and \(G_{n1}^{R}\) and describe how the points of the gadgets \(G_{i}^{P}\) and \(G_{i}^{R}\) can be chosen when the points of the gadgets \(G_{i+1}^{P}\) and \(G_{i+1}^{R}\) are already chosen. For j∈[2], we choose \(A^{n1}_{R,j}=(0,1)\), \(B^{n1}_{R,j}=(0,0)\), \(C^{n1}_{R,j}=(0.7,0.1)\), and \(D^{n1}_{R,j}=(1.2,0.08)\). Furthermore, we choose \(A^{n1}_{P,1}=(2,1.8)\), \(B^{n1}_{P,1}=(3.3,2.8)\), \(C^{n1}_{P,1}=(1.3,1.4)\), \(D^{n1}_{P,1}=(1.5,0.9)\), \(A^{n1}_{P,2}=(0.7,1.6)\), \(B^{n1}_{P,2}=(1.5,1.2)\), \(C^{n1}_{P,2}=(1.9,1.5)\), and \(D^{n1}_{P,2}=(0.8,1.1)\).
 1.
Start with the coordinates specified for the points of gadgets \(G_{i+1}^{P}\) and \(G_{i+1}^{R}\).
 2.
Scale each coordinate by a factor of 7.7.
 3.
Translate the points by the vector (1.93,0.3).
Let us remark that this also implies Theorem 1 for the L _{∞} metric because distances with respect to the L _{∞} metric coincide with distances with respect to the Manhattan metric if one rotates all points by π/4 around the origin and scales every coordinate by \(1/\sqrt{2}\).
3.2.2 Embedding the Construction into General L _{ p } Metrics
 1.
Start with the coordinates specified for the points of gadgets \(G_{i+1}^{P}\) and \(G_{i+1}^{R}\).
 2.
Rotate these points around the origin by π.
 3.
Scale each coordinate by a factor of 7.8.
 4.
Translate the points by the vector (7.2,5.3).
4 Expected Number of 2Changes
We analyze the expected number of 2changes on random ddimensional Manhattan and Euclidean instances, for an arbitrary constant dimension d≥2. One possible approach for this is to analyze the improvement made by the smallest improving 2change: If the smallest improvement is not too small, then the number of improvements cannot be large. This approach yields polynomial bounds, but in our analysis, we consider not only a single step but certain pairs of steps. We show that the smallest improvement made by any such pair is typically much larger than the improvement made by a single step, which yields better bounds. Our approach is not restricted to pairs of steps. One could also consider sequences of steps of length k for any small enough k. In fact, for general ϕperturbed graphs with m edges, we consider sequences of length \(\sqrt{\log{m}}\) in [5]. The reason why we can analyze longer sequences for general graphs is that these inputs possess more randomness than ϕperturbed Manhattan and Euclidean instances because every edge length is a random variable that is independent of the other edge lengths. Hence, the analysis for general ϕperturbed graphs demonstrates the limits of our approach under optimal conditions. For Manhattan and Euclidean instances, the gain of considering longer sequences is small due to the dependencies between the edge lengths.
4.1 Manhattan Instances
In this section, we analyze the expected number of 2changes on ϕperturbed Manhattan instances. First we prove a weaker bound than the one in Theorem 2 in a slightly different model. In this model the position of a vertex v _{ i } is not chosen according to a density function f _{ i }:[0,1]^{ d }→[0,ϕ], but instead each of its d coordinates is chosen independently. To be more precise, for every j∈[d], there is a density function \(f_{i}^{j}\colon[0,1] \to[0,\phi]\) according to which the jth coordinate of v _{ i } is chosen.
The proof of this weaker bound illustrates our approach and reveals the problems one has to tackle in order to improve the upper bounds. It is solely based on an analysis of the smallest improvement made by any of the possible 2Opt steps. If with high probability every 2Opt step decreases the tour length by an inverse polynomial amount, then with high probability only polynomially many 2Opt steps are possible before a local optimum is reached. In fact, the probability that there exists a 2Opt step that decreases the tour length by less than an inverse polynomial amount is so small that (as we will see) even the expected number of possible 2Opt steps can be bounded polynomially.
Theorem 7
Starting with an arbitrary tour, the expected number of steps performed by 2Opt on ϕperturbed Manhattan instances with n vertices is O(n ^{6}⋅logn⋅ϕ) if the coordinates of every vertex are drawn independently.
Proof
For each of these linear combinations, the probability that it takes a value in the interval (0,ε] is bounded from above by εϕ. To see this, we distinguish between two cases: If all coefficients in the linear combination are zero then the probability that the linear combination takes a value in the interval (0,ε] is zero. If at least one coefficient is nonzero then we can apply the principle of deferred decisions (see, e.g., [14]). Let \(x_{i}^{j}\) be a variable that has a nonzero coefficient α and assume that all random variables except for \(x_{i}^{j}\) are already drawn. Then, in order for the linear combination to take a value in the interval (0,ε], the random variable \(x_{i}^{j}\) has to take a value in a fixed interval of length ε/α. As the density of \(x_{i}^{j}\) is bounded from above by ϕ and α is a nonzero integer, the probability of this event is at most εϕ.
Since Δ(S) can only take a value in the interval (0,ε] if one of the linear combinations takes a value in this interval, the probability of the event Δ(S)∈(0,ε] can be upper bounded by (4!)^{ d } εϕ.
The bound in Theorem 7 is only based on the smallest improvement Δ _{min} made by any of the 2Opt steps. Intuitively, this is too pessimistic since most of the steps performed by 2Opt yield a larger improvement than Δ _{min}. In particular, two consecutive steps yield an improvement of at least Δ _{min} plus the improvement \(\varDelta_{\min}'\) of the second smallest step. This observation alone, however, does not suffice to improve the bound substantially. Instead, we show in Lemma 8 that we can regroup the 2changes to pairs such that each pair of 2changes is linked by an edge, i.e., one edge added to the tour in the first 2change is removed from the tour in the second 2change. Then we analyze the smallest improvement made by any pair of linked 2Opt steps. Obviously, this improvement is at least \(\varDelta_{\min}+\varDelta_{\min}'\) but one can hope that it is much larger because it is unlikely that the 2change that yields the smallest improvement and the 2change that yields the second smallest improvement form a pair of linked steps. We show that this is indeed the case and use this result to prove the bound on the expected length of the longest path in the state graph of 2Opt on ϕperturbed Manhattan instances claimed in Theorem 2.
4.1.1 Construction of Pairs of Linked 2Changes
Consider an arbitrary sequence of length t of consecutive 2changes. The following lemma guarantees that the number of disjoint linked pairs of 2changes in every such sequence increases linearly with the length t.
Lemma 8
In every sequence of t consecutive 2changes, the number of disjoint pairs of 2changes that are linked by an edge, i.e., pairs such that there exists an edge added to the tour in the first 2change of the pair and removed from the tour in the second 2change of the pair, is at least t/3−n(n−1)/4.
Proof
If one 2change S _{ i } is processed, it excludes at most two other 2changes from being processed (S _{ j } and S _{ j′}). Hence, the number of pairs added to \(\mathcal{L}\) is at least t/3−n(n−1)/4 because there can be at most \(\lfloor \binom{n}{2}/2\rfloor=\lfloor n(n1)/4\rfloor\) steps S _{ i } for which neither j nor j′ is defined. □
 pairs of type 0: {v _{2},v _{4}}∩{v _{5},v _{6}}=0. This case is illustrated in Fig. 7.
 pairs of type 1: {v _{2},v _{4}}∩{v _{5},v _{6}}=1. We can assume w.l.o.g. that v _{2}∈{v _{5},v _{6}}. We have to distinguish between two subcases: (a) The edges {v _{1},v _{5}} and {v _{2},v _{3}} are added to the tour in the second step. (b) The edges {v _{1},v _{2}} and {v _{3},v _{5}} are added to the tour in the second step. These cases are illustrated in Fig. 8.

pairs of type 2: {v _{2},v _{4}}∩{v _{5},v _{6}}=2. The case v _{2}=v _{5} and v _{4}=v _{6} cannot appear as it would imply that in the first step the edges {v _{1},v _{2}} and {v _{3},v _{4}} are exchanged with the edges {v _{1},v _{3}} and {v _{2},v _{4}}, and that in the second step the edges {v _{1},v _{3}} and {v _{2},v _{4}} are again exchanged with the edges {v _{1},v _{2}} and {v _{3},v _{4}}. Hence, one of these 2changes cannot be improving, and for pairs of this type we must have v _{2}=v _{6} and v _{4}=v _{5}.
When distances are measured according to the Euclidean metric, pairs of type 2 result in vast dependencies and hence the probability that there exists a pair of this type in which both steps are improvements by at most ε with respect to the Euclidean metric cannot be bounded appropriately. In order to reduce the number of cases we have to consider and in order to prepare for the analysis of ϕperturbed Euclidean instances, we exclude pairs of type 2 from our probabilistic analysis by leaving out all pairs of type 2 when constructing the list \(\mathcal{L}\) in the proof of Lemma 8.
We only need to show that there are always enough pairs of type 0 or 1. Consider two steps S _{ i } and S _{ j } with i<j that form a pair of type 2. Assume that in step S _{ i } the edges {v _{1},v _{2}} and {v _{3},v _{4}} are replaced by the edges {v _{1},v _{3}} and {v _{2},v _{4}}, and that in step S _{ j } these edges are replaced by the edges {v _{1},v _{4}} and {v _{2},v _{3}}. Now consider the next step S _{ l } with l>j in which the edge {v _{1},v _{4}} is removed from the tour, if such a step exists, and the next step S _{ l′} with l′>j in which the edge {v _{2},v _{3}} is removed from the tour if such a step exists. Observe that neither (S _{ j },S _{ l }) nor (S _{ j },S _{ l′}) can be a pair of type 2 because otherwise the improvement of one of the steps S _{ i }, S _{ j }, and S _{ l }, or of one of the steps S _{ i }, S _{ j }, and S _{ l′}, respectively, must be negative. In particular, we must have l≠l′.
If we encounter a pair (S _{ i },S _{ j }) of type 2 in the construction of the list \(\mathcal{L}\), we mark step S _{ i } as being processed without adding a pair of 2changes to \(\mathcal{L}\) and without removing S _{ j } from the sequence of steps to be processed. Let x denote the number of pairs of type 2 that we encounter during the construction of the list \(\mathcal{L}\). Our argument above shows that the number of pairs of type 0 or 1 that are added to \(\mathcal{L}\) is at least x−n(n−1)/4. This implies t≥x+(x−n(n−1)/4) and x≤t/2+n(n−1)/8. Hence, the number of relevant steps reduces from t to t′=t−x≥t/2−n(n−1)/8. Using this estimate in Lemma 8 yields the following lemma.
Lemma 9
In every sequence of t consecutive 2changes the number of disjoint pairs of 2changes of type 0 or 1 is at least t/6−7n(n−1)/24.
4.1.2 Analysis of Pairs of Linked 2Changes
The following lemma gives a bound on the probability that there exists a pair of type 0 or 1 in which both steps are small improvements.
Lemma 10
In a ϕperturbed Manhattan instance with n vertices, the probability that there exists a pair of type 0 or type 1 in which both 2changes are improvements by at most ε is O(n ^{6}⋅ε ^{2}⋅ϕ ^{2}).
Proof
We divide the set of possible pairs of linear combinations \((\varDelta_{1}^{\sigma},\varDelta_{2}^{\sigma})\) into three classes. We say that a pair of linear combinations belongs to class A if at least one of the linear combinations equals 0, we say that it belongs to class B if \(\varDelta_{1}^{\sigma}=\varDelta_{2}^{\sigma}\), and we say that it belongs to class C if \(\varDelta_{1}^{\sigma}\) and \(\varDelta_{2}^{\sigma}\) are linearly independent. For tuples of orders σ that yield pairs from class A, the event \(\mathcal{A}^{\sigma}\) cannot occur because the value of at least one linear combination is 0. For tuples σ that yield pairs from class B, the event cannot occur either because either \(\varDelta _{1}^{\sigma}\) or \(\varDelta_{2}^{\sigma}=\varDelta_{1}^{\sigma}\) is at most 0. For tuples σ that yield pairs from class C, we can apply Lemma 20 from Appendix B, which shows that the probability of the event \(\mathcal{A}^{\sigma}\) is bounded from above by (εϕ)^{2}. Hence, we only need to show that every pair \((\varDelta_{1}^{\sigma},\varDelta_{2}^{\sigma})\) of linear combinations belongs either to class A, B, or C.
If, for one i∈[d], the pair \((X_{1}^{\sigma_{i},i},X_{2}^{\sigma _{i},i})\) of linear combinations belongs to class C, then also the pair \((\varDelta_{1}^{\sigma},\varDelta_{2}^{\sigma})\) belongs to class C because the sets of variables occurring in \(X_{j}^{\sigma_{i},i}\) and \(X_{j}^{\sigma_{i'},i'}\) are disjoint for i≠i′. If for all i∈[d] the pair of linear combinations \((X_{1}^{\sigma_{i},i},X_{2}^{\sigma_{i},i})\) belongs to class A or B, then also the pair \((\varDelta_{1}^{\sigma},\varDelta_{2}^{\sigma})\) belongs either to class A or B. Hence, the following lemma directly implies that \((\varDelta_{1}^{\sigma},\varDelta_{2}^{\sigma})\) belongs to one of the classes A, B, or C.
Lemma 11
For pairs of type 0 and for i∈[d], the pair of linear combinations \((X_{1}^{\sigma_{i},i},X_{2}^{\sigma_{i},i})\) belongs either to class A, B, or C.
Proof

\(X_{1}^{\sigma_{i},i}\) does not contain \(x^{2}_{i}\) or \(x_{i}^{4}\). If \(x^{3}_{i}\ge x_{i}^{4}\), it must be true that \(x^{2}_{i}\ge x_{i}^{4}\) in order for \(x_{i}^{4}\) to cancel out. Then, in order for \(x_{i}^{2}\) to cancel out, it must be true that \(x^{2}_{i}\ge x_{i}^{1}\). If \(x^{3}_{i}\le x_{i}^{4}\), it must be true that \(x^{2}_{i}\le x_{i}^{4}\) in order for \(x_{i}^{4}\) to cancel out. Then, in order for \(x_{i}^{2}\) to cancel out, it must be true that \(x^{2}_{i}\le x_{i}^{1}\).
Hence, either \(x^{3}_{i}\ge x_{i}^{4}\), \(x^{2}_{i}\ge x_{i}^{4}\), and \(x^{2}_{i}\ge x_{i}^{1}\), or \(x^{3}_{i}\le x_{i}^{4}\), \(x^{2}_{i}\le x_{i}^{4}\), and \(x^{2}_{i}\le x_{i}^{1}\).

\(X_{2}^{\sigma_{i},i}\) does not contain \(x^{5}_{i}\) or \(x^{6}_{i}\). If \(x^{5}_{i}\ge x^{6}_{i}\), it must be true that \(x^{3}_{i}\ge x^{6}_{i}\) in order for \(x^{6}_{i}\) to cancel out, and it must be true that \(x^{5}_{i}\ge x_{i}^{1}\) in order for \(x^{5}_{i}\) to cancel out. If \(x^{5}_{i}\le x^{6}_{i}\), it must be true that \(x^{3}_{i}\le x^{6}_{i}\) in order for \(x^{6}_{i}\) to cancel out, and it must be true that \(x^{5}_{i}\le x_{i}^{1}\) in order for \(x^{5}_{i}\) to cancel out.
Hence, either \(x^{5}_{i}\ge x^{6}_{i}\), \(x^{3}_{i}\ge x^{6}_{i}\), and \(x^{5}_{i}\ge x_{i}^{1}\), or \(x^{5}_{i}\le x^{6}_{i}\), \(x^{3}_{i}\le x^{6}_{i}\), and \(x^{5}_{i}\le x_{i}^{1}\).
 \(x_{i}^{1}\ge x_{i}^{3}\):
 In this case, we can write \(X_{1}^{\sigma _{i},i}\) as Since we have argued above that either \(x^{3}_{i}\ge x_{i}^{4}\), \(x^{2}_{i}\ge x_{i}^{4}\), and \(x^{2}_{i}\ge x_{i}^{1}\), or \(x^{3}_{i}\le x_{i}^{4}\), \(x^{2}_{i}\le x_{i}^{4}\), and \(x^{2}_{i}\le x_{i}^{1}\), we obtain that eitheror$$X_1^{\sigma_i,i} = \bigl(x^2_ix^1_i \bigr)+\bigl(x^3_ix^4_i\bigr) \bigl(x^1_ix^3_i\bigr) \bigl(x^2_ix^4_i\bigr) = 2x^1_i+2x^3_i $$We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{5}_{i}\ge x^{6}_{i}\), \(x^{3}_{i}\ge x^{6}_{i}\), and \(x^{5}_{i}\ge x_{i}^{1}\), or \(x^{5}_{i}\le x^{6}_{i}\), \(x^{3}_{i}\le x^{6}_{i}\), and \(x^{5}_{i}\le x_{i}^{1}\), we obtain that either$$X_1^{\sigma_i,i} = \bigl(x^1_ix^2_i \bigr)+\bigl(x^4_ix^3_i\bigr) \bigl(x^1_ix^3_i\bigr) \bigl(x^4_ix^2_i\bigr) = 0. $$or$$X_2^{\sigma_i,i} = \bigl(x^1_ix^3_i \bigr)+\bigl(x^5_ix^6_i\bigr) \bigl(x^5_ix^1_i\bigr) \bigl(x^3_ix^6_i\bigr) = 2x^1_i2x^3_i $$In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x_{i}^{1}+2x^{3}_{i}\}\) and \(X_{2}^{\sigma_{i},i} \in\{0,2x_{i}^{1}2x^{3}_{i}\}\). Hence, in this case the resulting pair of linear combinations belongs either to class A or B.$$X_2^{\sigma_i,i} = \bigl(x^1_ix^3_i \bigr)+\bigl(x^6_ix^5_i\bigr) \bigl(x^1_ix^5_i\bigr) \bigl(x^6_ix^3_i\bigr) = 0. $$
 \(x_{i}^{3}\ge x_{i}^{1}\):
 In this case, we can write \(X_{1}^{\sigma _{i},i}\) as Since we have argued above that either \(x^{3}_{i}\ge x_{i}^{4}\), \(x^{2}_{i}\ge x_{i}^{4}\), and \(x^{2}_{i}\ge x_{i}^{1}\), or \(x^{3}_{i}\le x_{i}^{4}\), \(x^{2}_{i}\le x_{i}^{4}\), and \(x^{2}_{i}\le x_{i}^{1}\), we obtain that eitheror$$X_1^{\sigma_i,i} = \bigl(x^2_ix^1_i \bigr)+\bigl(x^3_ix^4_i\bigr) \bigl(x^3_ix^1_i\bigr) \bigl(x^2_ix^4_i\bigr) = 0 $$We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{5}_{i}\ge x^{6}_{i}\), \(x^{3}_{i}\ge x^{6}_{i}\), and \(x^{5}_{i}\ge x_{i}^{1}\), or \(x^{5}_{i}\le x^{6}_{i}\), \(x^{3}_{i}\le x^{6}_{i}\), and \(x^{5}_{i}\le x_{i}^{1}\), we obtain that either$$X_1^{\sigma_i,i} = \bigl(x^1_ix^2_i \bigr)+\bigl(x^4_ix^3_i\bigr) \bigl(x^3_ix^1_i\bigr) \bigl(x^4_ix^2_i\bigr) = 2x^1_i2x^3_i. $$or$$X_2^{\sigma_i,i} = \bigl(x^3_ix^1_i \bigr)+\bigl(x^5_ix^6_i\bigr) \bigl(x^5_ix^1_i\bigr) \bigl(x^3_ix^6_i\bigr) = 0 $$In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x^{1}_{i}2x^{3}_{i}\}\) and \(X_{2}^{\sigma_{i},i} \in\{0,2x^{1}_{i}+2x^{3}_{i}\}\). Hence, also in this case the resulting pair of linear combinations belongs either to class A or B. □$$X_2^{\sigma_i,i} = \bigl(x^3_ix^1_i \bigr)+\bigl(x^6_ix^5_i\bigr) \bigl(x^1_ix^5_i\bigr) \bigl(x^6_ix^3_i\bigr) = 2x^1_i+2x^3_i. $$
Lemma 12
For pairs of type 1(a) and for i∈[d], the pair \((X_{1}^{\sigma _{i},i},X_{2}^{\sigma_{i},i})\) of linear combinations belongs either to class A, B, or C.
Proof

\(X_{1}^{\sigma_{i},i}\) does not contain \(x_{i}^{4}\). If \(x^{3}_{i}\ge x_{i}^{4}\), it must be true that \(x^{2}_{i}\ge x_{i}^{4}\) in order for \(x_{i}^{4}\) to cancel out. If \(x^{3}_{i}\le x_{i}^{4}\), it must be true that \(x^{2}_{i}\le x_{i}^{4}\) in order for \(x_{i}^{4}\) to cancel out.
Hence, either \(x^{3}_{i}\ge x^{4}_{i}\) and \(x^{2}_{i}\ge x^{4}_{i}\), or \(x^{3}_{i}\le x^{4}_{i}\) and \(x^{2}_{i}\le x^{4}_{i}\).

\(X_{2}^{\sigma_{i},i}\) does not contain \(x^{5}_{i}\). If \(x^{2}_{i}\ge x^{5}_{i}\), it must be true that \(x^{1}_{i}\ge x_{i}^{5}\) in order for \(x^{5}_{i}\) to cancel out. If \(x^{2}_{i}\le x^{5}_{i}\), it must be true that \(x^{1}_{i}\le x_{i}^{5}\) in order for \(x^{5}_{i}\) to cancel out.
Hence, either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{1}\ge x^{5}_{i}\), or \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{1}\le x^{5}_{i}\).
 \(x_{i}^{1}\ge x_{i}^{3}\):
 In this case, we can write \(X_{1}^{\sigma _{i},i}\) as Since we have argued above that either \(x^{3}_{i}\ge x^{4}_{i}\) and \(x^{2}_{i}\ge x^{4}_{i}\), or \(x^{3}_{i}\le x^{4}_{i}\) and \(x^{2}_{i}\le x^{4}_{i}\), we obtain that either or We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{1}\ge x^{5}_{i}\), or if \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{1}\le x^{5}_{i}\), we obtain that either or In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x_{i}^{1}+2x_{i}^{2}, 2x_{i}^{1}+2x_{i}^{3},2x_{i}^{2}+2x_{i}^{3}\}\) and \(X_{2}^{\sigma_{i},i} \in\{0,2x^{1}_{i}2x^{2}_{i},2x^{1}_{i}2x^{3}_{i}, 2x^{2}_{i}2x^{3}_{i}\} \). Hence, in this case the resulting pair of linear combinations belongs either to class A, B, or C.
 \(x_{i}^{1}\le x_{i}^{3}\):
 In this case, we can write \(X_{1}^{\sigma _{i},i}\) as Since we have argued above that either \(x^{3}_{i}\ge x^{4}_{i}\) and \(x^{2}_{i}\ge x^{4}_{i}\), or \(x^{3}_{i}\le x^{4}_{i}\) and \(x^{2}_{i}\le x^{4}_{i}\), we obtain that either or We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{1}\ge x^{5}_{i}\), or \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{1}\le x^{5}_{i}\), we obtain that either or In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x^{1}_{i}2x^{2}_{i}, 2x^{1}_{i}2x^{3}_{i},2x_{i}^{2}2x_{i}^{3}\}\) and \(X_{2}^{\sigma_{i},i} \in\{ 0,2x^{1}_{i}+2x^{2}_{i},2x^{1}_{i}+2x^{3}_{i}, 2x^{2}_{i}+2x^{3}_{i}\}\). Hence, in this case the resulting pair of linear combinations belongs either to class A, B, or C. □
Lemma 13
For pairs of type 1(b) and for i∈[d], the pair of linear combinations \((X_{1}^{\sigma_{i},i},X_{2}^{\sigma_{i},i})\) belongs either to class A, B, or C.
Proof

\(X_{1}^{\sigma_{i},i}\) does not contain \(x_{i}^{4}\). We have considered this condition already for pairs of type 1(a) and showed that either \(x^{3}_{i}\ge x^{4}_{i}\) and \(x^{2}_{i}\ge x^{4}_{i}\), or \(x^{3}_{i}\le x^{4}_{i}\) and \(x^{2}_{i}\le x^{4}_{i}\).

\(X_{2}^{\sigma_{i},i}\) does not contain \(x^{5}_{i}\). If \(x^{2}_{i}\ge x^{5}_{i}\), it must be true that \(x^{3}_{i}\ge x_{i}^{5}\) in order for \(x^{5}_{i}\) to cancel out. If \(x^{2}_{i}\le x^{5}_{i}\), it must be true that \(x^{3}_{i}\le x_{i}^{5}\) in order for \(x^{5}_{i}\) to cancel out.
Hence, either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{3}\ge x^{5}_{i}\), or \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{3}\le x^{5}_{i}\).
 \(x_{i}^{1}\ge x_{i}^{3}\):

We have argued already for pairs of type 1(a) that in this case \(X_{1}^{\sigma_{i},i} \in\{ 0,2x_{i}^{1}+2x_{i}^{2},2x_{i}^{1}+2x_{i}^{3},2x_{i}^{2}+2x_{i}^{3}\}\).
We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{3}\ge x^{5}_{i}\), or \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{3}\le x^{5}_{i}\), we obtain that either or In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x_{i}^{1}+2x_{i}^{2}, 2x_{i}^{1}+2x_{i}^{3},2x_{i}^{2}+2x_{i}^{3}\}\) and \(X_{2}^{\sigma_{i},i} \in\{0,2x_{i}^{1}2x_{i}^{2},2x_{i}^{1}2x_{i}^{3}, 2x_{i}^{2}2x_{i}^{3}\} \). Hence, in this case the resulting pair of linear combinations belongs either to class A, B, or C.  \(x_{i}^{1}\le x_{i}^{3}\):

We have argued already for pairs of type 1(a) that in this case \(X_{1}^{\sigma_{i},i} \in\{0,2x^{1}_{i}2x^{2}_{i},2x^{1}_{i}2x^{3}_{i},2x_{i}^{2}2x_{i}^{3}\}\).
We can write \(X_{2}^{\sigma_{i},i}\) as Since we have argued above that either \(x^{2}_{i}\ge x^{5}_{i}\) and \(x_{i}^{3}\ge x^{5}_{i}\), or \(x^{2}_{i}\le x^{5}_{i}\) and \(x_{i}^{3}\le x^{5}_{i}\), we obtain that either or In summary, the case analysis shows that \(X_{1}^{\sigma_{i},i} \in\{ 0,2x^{1}_{i}2x^{2}_{i}, 2x^{1}_{i}2x^{3}_{i},2x_{i}^{2}2x_{i}^{3}\}\) and \(X_{2}^{\sigma_{i},i} \in\{ 0,2x^{1}_{i}+2x^{2}_{i},2x^{1}_{i}+2x^{3}_{i}, 2x^{2}_{i}+2x^{3}_{i}\}\). Hence, in this case the resulting pair of linear combinations belongs either to class A, B, or C. □
We have argued above that for tuples σ of orders that yield pairs from class A or B, the event \(\mathcal{A}^{\sigma}\) cannot occur. For tuples σ that yield pairs from class C, we can apply Lemma 20 from Appendix B, which shows that the probability of the event \(\mathcal{A}^{\sigma}\) is bounded from above by (εϕ)^{2}. As we have shown that every tuple yields a pair from class A, B, or C, we can conclude the proof of Lemma 10 by a union bound over all pairs of linked 2changes of type 0 and 1 and all tuples σ. As these are O(n ^{6}), the lemma follows. □
4.1.3 Expected Number of 2Changes
Based on Lemmas 9 and 10, we are now able to prove part (a) of Theorem 2.
Proof of Theorem 2(a)
This concludes the proof of part (a) of the theorem. □
Chandra, Karloff, and Tovey [3] show that for every metric that is induced by a norm on \(\mathbb{R}^{d}\), and for any set of n points in the unit hypercube [0,1]^{ d }, the optimal tour visiting all n points has length O(n ^{(d−1)/d }). Furthermore, every insertion heuristic finds an O(logn)approximation [18]. Hence, if one starts with a solution computed by an insertion heuristic, the initial tour has length O(n ^{(d−1)/d }⋅logn). Using this observation yields part (a) of Theorem 3:
Proof of Theorem 3(a)
4.2 Euclidean Instances
In this section, we analyze the expected number of 2changes on ϕperturbed Euclidean instances. The analysis is similar to the analysis of Manhattan instances in the previous section; only Lemma 10 needs to be replaced by the following equivalent version for the L _{2} metric, which will be proved later in this section.
Lemma 14
For ϕperturbed L _{2} instances, the probability that there exists a pair of type 0 or type 1 in which both 2changes are improvements by at most ε≤1/2 is bounded by O(n ^{6}⋅ϕ ^{5}⋅ε ^{2}⋅log^{2}(1/ε))+O(n ^{5}⋅ϕ ^{4}⋅ε ^{3/2}⋅log(1/ε)).
The bound that this lemma provides is slightly weaker than its L _{1} counterpart, and hence also the bound on the expected running time is slightly worse for L _{2} instances. The crucial step to proving Lemma 14 is to gain a better understanding of the random variable that describes the improvement of a single fixed 2change. In the next section, we analyze this random variable under several conditions, e.g., under the condition that the length of one of the involved edges is fixed. With the help of these results, pairs of linked 2changes can easily be analyzed. Let us mention that our analysis of a single 2change yields a bound of O(n ^{7}⋅log^{2}(n)⋅ϕ ^{3}) for the expected number of 2changes. For Euclidean instances in which all points are distributed uniformly at random over the unit square, this bound already improves the best previously known bound of O(n ^{10}⋅logn).
4.2.1 Analysis of a Single 2Change
We analyze a 2change in which the edges {O,Q _{1}} and {P,Q _{2}} are exchanged with the edges {O,Q _{2}} and {P,Q _{1}} for some vertices O, P, Q _{1}, and Q _{2}. In the input model we consider, each of these points has a probability distribution over the unit hypercube according to which it is chosen. In this section, we consider a simplified random experiment in which O is chosen to be the origin and P, Q _{1}, and Q _{2} are chosen independently and uniformly at random from a ddimensional hyperball with radius \(\sqrt{d}\) centered at the origin. In the next section, we argue that the analysis of this simplified random experiment helps to analyze the actual random experiment that occurs in the probabilistic input model.
Due to the rotational symmetry of the simplified model, we assume without loss of generality that P lies at position (0^{ d−1},T) for some T≥0. For i∈[2], Let Z _{ i } denote the difference d(O,Q _{ i })−d(P,Q _{ i }). Then the improvement Δ of the 2change can be expressed as Z _{1}−Z _{2}. The random variables Z _{1} and Z _{2} are identically distributed, and they are independent if T is fixed. We denote by \(f_{Z_{1}\mid T=\tau,R=r}\) the density of Z _{1} conditioning on the fact that d(O,Q _{1})=r and T=τ. Similarly, we denote by \(f_{Z_{2}\mid T=\tau,R=r}\) the density of Z _{2} conditioning on the fact that d(O,Q _{2})=r and T=τ. As Z _{1} and Z _{2} are identically distributed, the conditional densities \(f_{Z_{1}\mid T=\tau,R=r}\) and \(f_{Z_{2}\mid T=\tau,R=r}\) are identical as well. Hence, we can drop the index in the following and write f _{ Z∣T=τ,R=r }.
Lemma 15
Proof
First case: r≥τ.
Second case: r<τ.
Based on Lemma 15, the density of the random variable Δ=Z _{1}−Z _{2} under the conditions R _{1}:=d(O,Q _{1})=r _{1}, R _{2}:=d(O,Q _{2})=r _{2}, and T:=d(O,P)=τ can be computed as the convolution of the densities of the random variables Z _{1} and −Z _{2}. The former density equals f _{ Z∣T=τ,R=r } and the latter density can easily be obtained from f _{ Z∣T=τ,R=r }.
Lemma 16
The simple but somewhat tedious calculation that yields Lemma 16 is deferred to Appendix C.1. In order to prove Lemma 14, we need bounds on the densities of the random variables Δ, Z _{1}, and Z _{2} under certain conditions. We summarize these bounds in the following lemma.
Lemma 17
 (a)For i∈[2], the density of Δ under the condition R _{ i }=r is bounded by$$f_{\varDelta\mid R_i=r}(\delta) \le\frac{\kappa}{\sqrt{r}}\cdot\ln \bigl( \delta^{1} \bigr). $$
 (b)The density of Δ under the condition T=τ is bounded by$$f_{\varDelta\mid T=\tau}(\delta) \le\frac{\kappa}{\tau}\cdot\ln \bigl( \delta^{1} \bigr). $$
 (c)The density of Δ is bounded by$$f_{\varDelta}(\delta) \le \kappa\cdot\ln \bigl(\delta^{1} \bigr). $$
 (d)For i∈[2], the density of Z _{ i } under the condition T=τ is bounded byif z<τ. Since Z _{ i } takes only values in the interval [−τ,τ], the conditional density \(f_{Z_{i}\mid T=\tau}(z)\) is 0 for z∉[−τ,τ].$$f_{Z_i\mid T=\tau}(z) \le\frac{\kappa}{\sqrt{\tau^2z^2}} $$
Lemma 17 follows from Lemmas 15 and 16 by integrating over all values of the unconditioned distances. The proof can be found in Appendix C.2.
4.2.2 Simplified Random Experiments
In the previous section we did not analyze the random experiment that really takes place. Instead of choosing the points according to the given density functions, we simplified their distributions by placing point O in the origin and by giving the other points P, Q _{1}, and Q _{2} uniform distributions centered around the origin. In our input model, however, each of these points is described by a density function over the unit hypercube. We consider the probability of the event Δ∈[0,ε] in the original input model as well as in the simplified random experiment. In the following, we denote this event by \(\mathcal{E}\). We claim that the simplified random experiment that we analyze is only slightly dominated by the original random experiment, in the sense that the probability of the event \(\mathcal{E}\) in the simplified random experiment is smaller by at most some factor depending on ϕ.
Taking into account this factor and using Lemma 17(c) and a union bound over all possible 2changes yields the following lemma about the improvement of a single 2change.
Lemma 18
The probability that there exists an improving 2change whose improvement is at most ε≤1/2 is bounded from above by O(n ^{4}⋅ϕ ^{3}⋅ε⋅log(1/ε)).
Proof
Using similar arguments as in the proof of Theorem 7 yields the following upper bound on the expected number of 2changes.
Theorem 19
Starting with an arbitrary tour, the expected number of steps performed by 2Opt on ϕperturbed Euclidean instances is O(n ^{7}⋅log^{2}(n)⋅ϕ ^{3}).
Proof
Pairs of Type 0
In order to improve upon Theorem 19, we consider pairs of linked 2changes as in the analysis of ϕperturbed Manhattan instances. Since our analysis of pairs of linked 2changes is based on the analysis of a single 2change that we presented in the previous section, we also have to consider simplified random experiments when analyzing pairs of 2changes. For a fixed pair of type 0, we assume that point v _{3} is chosen to be the origin and the other points v _{1}, v _{2}, v _{4}, v _{5}, and v _{6} are chosen uniformly at random from a hyperball with radius \(\sqrt{d}\) centered at v _{3}. Let \(\mathcal{E}\) denote the event that both Δ _{1} and Δ _{2} lie in the interval [0,ε], for some given ε. With the same arguments as above, one can see that the probability of \(\mathcal{E}\) in the simplified random experiment is smaller compared to the original experiment by at most a factor of ((4d)^{ d/2} ϕ)^{5}. The exponent 5 is due to the fact that we have now five other points instead of only three.
Pairs of Type 1
For a fixed pair of type 1, we consider the simplified random experiment in which v _{2} is placed in the origin and the other points v _{1}, v _{3}, v _{4}, and v _{5} are chosen uniformly at random from a hyperball with radius \(\sqrt{d}\) centered at v _{2}. In this case, the probability in the simplified random experiment is smaller by at most a factor of ((4d)^{ d/2} ϕ)^{4}. The exponent 4 is due to the fact that we have now four other points.
4.2.3 Analysis of Pairs of Linked 2Changes
Finally, we can prove Lemma 14.
Proof of Lemma 14
We start by considering pairs of type 0. We consider the simplified random experiment in which v _{3} is chosen to be the origin and the other points are drawn uniformly at random from a hyperball with radius \(\sqrt{d}\) centered at v _{3}. If the position of the point v _{1} is fixed, then the events Δ _{1}∈[0,ε] and Δ _{2}∈[0,ε] are independent as only the vertices v _{1} and v _{3} appear in both the first and the second step. In fact, because the densities of the points v _{2}, v _{4}, v _{5}, and v _{6} are rotationally symmetric, the concrete position of v _{1} is not important in our simplified random experiment anymore; only the distance R between v _{1} and v _{3} is of interest.
It remains to consider pairs of type 1. We consider the simplified random experiment in which v _{2} is chosen to be the origin and the other points are drawn uniformly at random from a hyperball with radius \(\sqrt{d}\) centered at v _{2}. In contrast to pairs of type 0, pairs of type 1 exhibit larger dependencies as only 5 different vertices are involved in these pairs. Fix one pair of type 1. The two 2changes share the whole triangle consisting of v _{1}, v _{2}, and v _{3}. In the second step, there is only one new vertex, namely v _{5}. Hence, there is not enough randomness contained in a pair of type 1 such that Δ _{1} and Δ _{2} are nearly independent as for pairs of type 0.
We start by considering pairs of type 1(a) as defined in Sect. 4.1.1. First, we analyze the probability that Δ _{1} lies in the interval [0,ε]. After that, we analyze the probability that Δ _{2} lies in the interval [0,ε] under the condition that the points v _{1}, v _{2}, v _{3}, and v _{4} have already been chosen. In the analysis of the second step we cannot make use of the fact that the distances d(v _{1},v _{3}) and d(v _{2},v _{3}) are random variables anymore since we exploited their randomness already in the analysis of the first step. The only distances whose randomness we can exploit are the distances d(v _{1},v _{5}) and d(v _{2},v _{5}). We pessimistically assume that the distances d(v _{1},v _{3}) and d(v _{2},v _{3}) have been chosen by an adversary. This means the adversary can determine an interval of length ε in which the random variable d(v _{2},v _{5})−d(v _{1},v _{5}) must lie in order for Δ _{2} to lie in [0,ε].
4.2.4 The Expected Number of 2Changes
Based on Lemmas 9 and 14, we are now able to prove part (b) of Theorem 2, which states that the expected length of the longest path in the 2Opt state graph is O(n ^{4+1/3}⋅log(nϕ)⋅ϕ ^{8/3}) for ϕperturbed Euclidean instances with n points.
Proof of Theorem 2(b)
Using the same observations as in the proof of Theorem 3(a) also yields part (b):
Proof of Theorem 3(b)
Estimating the length of the initial tour by O(n ^{(d−1)/d }⋅logn) instead of O(n) improves the upper bound on the expected number of 2changes by a factor of Θ(n ^{1/d }/logn) compared to Theorem 2(b). This observation yields the bound claimed in Theorem 3(b). □
5 Expected Approximation Ratio
In this section, we consider the expected approximation ratio of the solution found by 2Opt on ϕperturbed L _{ p } instances. Chandra, Karloff, and Tovey [3] show that if one has a set of n points in the unit hypercube [0,1]^{ d } and the distances are measured according to a metric that is induced by a norm, then every locally optimal solution has length at most c⋅n ^{(d−1)/d } for an appropriate constant c depending on the dimension d and the metric. Hence, it follows for every L _{ p } metric that 2Opt yields a tour of length O(n ^{(d−1)/d }) on ϕperturbed L _{ p } instances. This implies that the approximation ratio of 2Opt on these instances can be bounded from above by \(O(n^{(d1)/d})/\operatorname{Opt}\), where \(\operatorname{Opt}\) denotes the length of the shortest tour. We will show a lower bound on \(\operatorname{Opt}\) that holds with high probability in ϕperturbed L _{ p } instances. Based on this, we prove Theorem 4.
Proof of Theorem 4
Next we show that X is sharply concentrated around its mean value. The random variable X is the sum of k 01random variables X _{ i }. If these random variables were independent, we could simply use a Chernoff bound to bound the probability that X takes a value that is much smaller than its mean value. Intuitively, whenever we already know that some of the X _{ i }’s are zero, then the probability of the event that another X _{ i } also takes the value zero becomes smaller. Hence, intuitively, the dependencies can only help to bound the probability that X takes a value smaller than its mean value.
 If \(X\ge\frac{n}{2^{d+3}}\), then, assuming that n is large enough, we have that X>3^{ d } and hence, (5.1) implies thatwhere we used that k=Θ(nϕ) for the last equation. Combining this with Chandra, Karloff, and Tovey’s [3] result that every locally optimal solution has length at most O(n ^{(d−1)/d }) yields an approximation ratio of$$\operatorname{Opt}\ge \biggl\lceil\frac{X}{3^d} \biggr\rceil\cdot \frac{1}{\sqrt[d]{k}} \ge \frac{X}{3^d\sqrt[d]{k}} \ge\frac{n}{2^{d+3}3^d\sqrt[d]{k}} = \varTheta \biggl(\frac{n^{(d1)/d}}{\sqrt[d]{\phi}} \biggr), $$$$\frac{O(n^{(d1)/d})}{\varTheta (\frac{n^{(d1)/d}}{\sqrt [d]{\phi }} )} = O\bigl(\sqrt[d]{\phi}\bigr). $$

If \(X < \frac{n}{2^{d+3}}\), then we use n as an upper bound on the approximation ratio of any locally optimal solution. This bound holds in fact for any possible tour, as the following argument shows: The length of every tour is bounded from above by n times the length α of the longest edge. Let u and v be the vertices that this edge connects. Then every tour has to contain a path between u and v. Due to the triangle inequality, this path must have length at least α.
We have seen in (5.2) that the event \(X < \frac{n}{2^{d+3}}\) occurs only with exponentially small probability. This implies that it adds at mostto the expected approximation ratio.$$\exp \biggl(\frac{n}{2^{d+5}} \biggr)\cdot n = o(1) $$
6 Smoothed Analysis
Smoothed Analysis was introduced by Spielman and Teng [19] as a hybrid of worst case and average case analysis. The semirandom input model in a smoothed analysis is designed to capture the behavior of algorithms on typical inputs better than a worst case or average case analysis alone as it allows an adversary to specify an arbitrary input which is randomly perturbed afterwards. In Spielman and Teng’s analysis of the Simplex algorithm the adversary specifies an arbitrary linear program which is perturbed by adding independent Gaussian random variables to each number in the linear program. Our probabilistic analysis of Manhattan and Euclidean instances can also be seen as a smoothed analysis in which an adversary can choose the distributions for the points over the unit hypercube. The adversary is restricted to distributions that can be represented by densities that are bounded by ϕ. Our model cannot handle Gaussian perturbations directly because the support of Gaussian random variables is not bounded.
Assume that every point v _{1},…,v _{ n } is described by a density whose support is restricted to the hypercube [−α,1+α]^{ d }, for some α≥1. Then after appropriate scaling and translating, we can assume that all supports are restricted to the unit hypercube [0,1]^{ d }. Thereby, the maximal density ϕ increases by at most a factor of (2α+1)^{ d }. Hence, after appropriate scaling and translating, Theorems 2, 3, and 4 can still be applied if one takes into account the increased densities.
7 Conclusions and Open Problems
We have shown several new results on the running time and the approximation ratio of the 2Opt heuristic. However, there are still a variety of open problems regarding this algorithm. Our lower bounds only show that there exist families of instances on which 2Opt takes an exponential number of steps if it uses a particular pivot rule. It would be interesting to analyze the diameter of the state graph and to either present instances on which every pivot rule needs an exponential number of steps or to prove that there is always an improvement sequence of polynomial length to a locally optimal solution. Also the worst number of local improvements for some natural pivot rules like, e.g., the one that always makes the largest possible improvement or the one that always chooses a random improving 2change, is not known yet. Furthermore, the complexity of computing locally optimal solutions is open. The only result in this regard is due to Krentel [9] who shows that it is PLScomplete to compute a local optimum for the metric TSP for kOpt for some constant k. It is not known whether his construction can be embedded into the Euclidean metric and whether it is PLScomplete to compute locally optimal solutions for 2Opt. Fischer and Torenvliet [6] show, however, that for the general TSP, it is PSPACEhard to compute a local optimum for 2Opt that is reachable from a given initial tour.
The obvious open question concerning the probabilistic analysis is how the gap between experiments and theory can be narrowed further. In order to tackle this question, new methods seem to be necessary. Our approach, which is solely based on analyzing the smallest improvement made by a sequence of linked 2changes, seems to yield too pessimistic bounds. Another interesting area to explore is the expected approximation ratio of 2Opt. In experiments, approximation ratios close to 1 are observed. For instances that are chosen uniformly at random, the bound on the expected approximation ratio is a constant but unfortunately a large one. It seems to be a very challenging problem to improve this constant to a value that matches the experimental results.
Besides 2Opt, there are also other local search algorithms that are successful for the traveling salesperson problem. In particular, the Lin–Kernighan heuristic [11] is one of the most successful local search algorithm for the symmetric TSP. It is a variant of kOpt in which k is not fixed and it can roughly be described as follows: Each local modification starts by removing one edge {a,b} from the current tour, which results in a Hamiltonian path with the two endpoints a and b. Then an edge {b,c} is added, which forms a cycle; there is a unique edge {c,d} incident to c whose removal breaks the cycle, producing a new Hamiltonian path with endpoints a and d. This operation is called a rotation. Now either a new Hamiltonian cycle can be obtained by adding the edge {a,d} to the tour or another rotation can be performed. There are a lot of different variants and heuristic improvements of this basic scheme, but little is known theoretically. Papadimitriou [16] shows for a variant of the Lin–Kernighan heuristic that computing a local optimum is PLScomplete, which is a sharp contrast to the experimental results. Since the Lin–Kernighan heuristic is widely used in practice, a theoretical explanation for its good behavior in practice is of great interest. Our analysis of 2Opt relies crucially on the fact that there are only a polynomial number of different 2changes. For the Lin–Kernighan heuristic, however, the number of different local improvements is exponential. Hence, it is an interesting question as to whether nonetheless the smallest possible improvement is polynomially large or whether different methods yield a polynomial upper bound on the expected running time of the Lin–Kernighan heuristic.
Notes
Acknowledgements
This work was supported in part by the EU within the 6th Framework Programme under contract 001907 (DELIS), by DFG grants VO 889/2 and WE 2842/1, and by EPSRC grant EP/F043333/1. We thank the referee for extraordinary efforts and many helpful suggestions.
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