Searching the k-change neighborhood for TSP is W[1]-hard

doi:10.1016/j.orl.2007.02.008

Operations Research Letters

Volume 36, Issue 1, January 2008, Pages 31-36

https://doi.org/10.1016/j.orl.2007.02.008 Get rights and content

Abstract

We show that searching the k-change neighborhood is W[1]-hard for metric TSP, which means that finding the best tour in the k-change neighborhood essentially requires complete search (modulo some complexity-theoretic assumptions).

Introduction

The Traveling Salesperson Problem (TSP) is one of the most studied optimization problems. Given the NP-hardness of this practically important problem, many heuristic approaches were proposed in the literature (cf., [8], [9]). Many of these algorithms are based on local search, which means that the algorithm produces better and better solutions by iteratively improving the current best solution. The improvement is local: only those tours are considered that are “close” to the current tour, that is, those tours that can be reached by applying certain operations on the current tour. The effectiveness of this search procedure will largely depend on the set of allowed operations that is used to find the improved tour. Presumably, having a larger set of operations increases our chances of finding a better tour, but the time required to search the local neighborhood increases.

Perhaps the most studied local search heuristic is the k-change neighborhood rule, where those tours are considered that can be reached from the current tour by replacing at most k arcs. Computational experiments are mostly done for $k = 2, 3,$ or 4. For larger k, however, the method becomes impractical, as there are $n^{O (k)}$ possible tours that have to be considered for improvement. A brute force search of the k-change neighborhood is not feasible for large k; therefore, unless we have a more efficient algorithm for finding the best tour in the local neighborhood, local search based on the k-change neighborhood cannot be implemented efficiently. On the other hand, an algorithm with running time of the form $O (2^{k} \cdot n)$ would make it possible to search the k-change neighborhood for, say, $k = 10$ .

The hardness of searching the k-change neighborhood can be studied very naturally in the framework of parameterized complexity, as suggested in [3]. In classical complexity theory, the hardness of a problem is usually analyzed as a function of the input size. Parameterized complexity studies the hardness of a problem in finer detail: certain parameters of the problem instances are defined, and the running time is expressed not only as a function of the input size, but also as a function of these parameters. For example, in classical complexity, the problems MAXIMUM CLIQUE and MINIMUM VERTEX COVER have the same complexity: both are NP-hard. In the decision version of these problems, each instance has a very natural parameter: the size k of the clique/vertex cover that we are looking for. Both problems can be solved in $n^{O (k)}$ time by complete enumeration. This means that the problems are polynomial-time solvable for every fixed value of k, but the algorithms become practically useless for large values of n, even if k is as small as 10. The problem is that k appears in the exponent of n, which means that the polynomial degree increases as k increases. However, MINIMUM VERTEX COVER has better algorithms, for example it can be solved in $O (2^{k} \cdot n^{2})$ time. It follows that MINIMUM VERTEX COVER has a quadratic algorithm for every fixed value of k. The central issue of parameterized complexity is the question whether it is possible to find algorithms where the exponent of the input size does not increase as parameter k increases. We say that a parameterized problem is fixed-parameter tractable if it can be solved in time $f (k) \cdot n^{c}$ , where c is a constant and f is a (possibly exponential) function depending only on k. This means that the exponential increase of the running time can be restricted to the parameter k. It turns out that several NP-hard problems are fixed-parameter tractable, for example MINIMUM VERTEX COVER, LONGEST PATH, and DISJOINT TRIANGLES. Analogously to NP-completeness in classical complexity, the theory of W[1]-hardness can be used to show that a problem is unlikely to be fixed-parameter tractable, i.e., for every algorithm the parameter has to appear in the exponent of n.

Returning to the TSP problem, what we would like to have is an algorithm that, given a tour T, finds the best tour in the k-neighborhood of T. If k is part of the input, then this problem is clearly NP-hard: if $k = n$ , then the problem is equivalent to finding the best possible tour. On the other hand, the problem is polynomial-time solvable for every fixed value of k (in $n^{O (k)}$ time by complete search). In this paper we investigate whether the brute force search can be improved to an algorithm where the degree of n does not depend on k, i.e., whether the problem is fixed-parameter tractable. The main result of the paper is that finding the best tour in the k-change neighborhood is W[1]-hard, which implies that the problem is not fixed-parameter tractable, unless $W [1] = FPT$ . This answers an open question of Fellows [3]. The hardness result holds even if the distance matrix is symmetric and satisfies the triangle inequality. However, our result does not say anything about the important special case when the cities are points in the plane, and the distance is Euclidean distance. It remains an interesting open question whether the hardness result can be extended to this case as well.

Section snippets

Parameterized complexity

We follow [5] for the standard definitions of parameterized complexity. Let $Σ$ be a finite alphabet. A decision problem is represented by a set $Q \subseteq Σ^{*}$ of strings over $Σ$ . A parameterization of a problem is a polynomial-time computable function $κ : Σ^{*} \to N$ . A parameterized decision problem is a pair $(Q, κ)$ , where $Q \subseteq Σ^{*}$ is an arbitrary decision problem and $κ$ is a parameterization. Intuitively, we can imagine a parameterized problem as a decision problem where each input instance $x \in Σ^{*}$ has a positive integer $κ$

Hardness result

The input of the Traveling Salesperson Problem (TSP) consists of a set V of cities and a distance matrix. The distance matrix contains a positive integer $d (u, v)$ for each ordered pair $(u, v)$ of cities. For convenience, we allow $d (u, v) = \infty$ . The task is to find a tour of minimal length that visits all the cities. That is, we have to find an ordering $v_{1},..., v_{n}$ of the cities such that the total length $\sum_{i = 1}^{n - 1} d (v_{i}, v_{i + 1}) + d (v_{n}, v_{1})$ is minimal (where $n = | V |$ ). A tour can also be considered as a set of n

Acknowledgment

I am grateful to Mike Fellows for directing my attention to the problem.

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