Lin-Kernighan Heuristic Adaptations for the Generalized Traveling Salesman Problem

arXiv:1003.5330v2 [cs.DS] 23 Jun 2010 Lin-Kernighan Heuristic Adaptations for the Generalized Traveling Salesman Problem D. Karapetyana,∗, G. Gutina aRoyal Holloway London University, Egham, Surrey, TW20 0EX, United Kingdom Abstract The Lin-Kernighan heuristic is known to be one of the most successful heuristics for the Traveling Salesman Problem (TSP). It has also proven its efficiency in application to some other problems. In this paper we discuss possible adaptations of TSP heuristics for the Generalized Traveling Salesman Problem (GTSP) and focus on the case of the Lin-Kernighan al- gorithm. At first, we provide an easy-to-understand description of the original Lin- Kernighan heuristic. Then we propose several adaptations, both trivial and complicated. Finally, we conduct a fair competition between all the variations of the Lin-Kernighan adaptation and some other GTSP heuristics. It appears that our adaptation of the Lin-Kernighan algorithm for the GTSP reproduces the success of the original heuristic. Different variations of our adaptation outperform all other heuristics in a wide range of trade-offs between solution quality and running time, making Lin-Kernighan the state-of-the-art GTSP local search. Keywords: Heuristics, Lin-Kernighan, Generalized Traveling Salesman Problem, Combinatorial Optimization. 1. Introduction One of the most successful heuristic algorithms for the famous Traveling Salesman Problem (TSP) known so far is the Lin-Kernighan heuristic (Lin and Kernighan, 1973). It was proposed almost forty years ago but even nowadays it is the state-of-the-art TSP local search (Johnson and McGeoch, 2002). In this paper we attempt to reproduce the success of the original TSP Lin-Kernighan heuristic for the Generalized Traveling Salesman Problem (GTSP), which is an important extension of TSP. In the TSP, we are given a set V of n vertices and weights w(x →y) of moving from a vertex x ∈V to a vertex y ∈V . A feasible solution, or a tour, is a cycle visiting every vertex in V exactly once. In the GTSP, we are given a set V of n vertices, weights w(x →y) of moving from x ∈V to y ∈V and a partition of V into m ∗Corresponding author Email addresses: daniel.karapetyan@gmail.com (D. Karapetyan), gutin@cs.rhul.ac.uk (G. Gutin) Preprint submitted to Elsevier November 26, 2024 nonempty clusters C1, C2, . . . , Cm such that Ci ∩Cj = ∅for each i ̸= j and S i Ci = V . A feasible solution, or a tour, is a cycle visiting exactly one vertex in every cluster. The objective of both TSP and GTSP is to find the shortest tour. If the weight matrix is symmetric, i.e., w(x →y) = w(y →x) for any x, y ∈V , the problem is called symmetric. Otherwise it is an asymmetric GTSP. In what follows, the number of vertices in cluster Ci is denoted as |Ci|, the size of the largest cluster is s, and Cluster(x) is the cluster containing a vertex x. The weight function w can be used for edges, paths w(x1 →x2 →. . . →xk) = w(x1 →x2)+w(x2 →x3)+. . .+w(xk−1 → xk), and cycles. Since Lin-Kernighan is designed for the symmetric problem, we do not consider the asymmetric GTSP in this research. However, some of the algorithms proposed in this paper are naturally suited for both symmetric and asymmetric cases. Observe that the TSP is a special case of the GTSP when |Ci| = 1 for each i and, hence, the GTSP is NP-hard. The GTSP has a host of applications in warehouse order picking with multiple stock locations, sequencing computer files, postal routing, airport selection and routing for courier planes and some others, see, e.g., (Fischetti et al., 1995, 1997; Laporte et al., 1996; Noon and Bean, 1991) and references therein. A lot of attention was paid in the literature to solving the GTSP. Several researchers (Ben-Arieh et al., 2003; Laporte and Semet, 1999; Noon and Bean, 1993) proposed trans- formations of the GTSP into the TSP. At first glance, the idea to transform a little-studied problem into a well-known one seems to be natural; however, this approach has a very lim- ited application. On the one hand, it requires exact solutions of the obtained TSP instances because even a near-optimal solution of such TSP may correspond to an infeasible GTSP solution. On the other hand, the produced TSP instances have quite an unusual structure which is difficult for the existing solvers. A more efficient way to solve the GTSP ex- actly is a branch-and-bound algorithm designed by Fischetti et al. (1997). This algorithm was able to solve instances with up to 89 clusters. Two approximation algorithms were proposed in the literature, but both of them are unsuitable for the general case of the prob- lem, and the guarantied solution quality is unreasonably low for real-world applications, see (Bontoux et al., 2010) and references therein. In order to obtain good (i.e., not necessarily exact) solutions for larger GTSP in- stances, one should use the heuristic approach. Several construction heuristics and local searches were discussed in (Bontoux et al., 2010; Gutin

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