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Hierarchical Solution of the Traveling Salesman Problem with

Random Dyadic Tilings

Tamás Kalmár-Nagy∗ and Bendegúz Dezső Bak†

Department of Fluid Mechanics, Faculty of Mechanical Engineering Budapest University of Technology and Economics

Abstract

We propose a hierarchical heuristic approach for solving the Traveling Salesman Problem (TSP) in the unit square. The points are partitioned with a random dyadic tiling and clusters are formed by the points located in the same tile. Each cluster is represented by its geometrical barycenter and a “coarse” TSP solution is calculated for these barycenters. Midpoints are placed at the middle of each edge in the coarse solution. Near-optimal (or optimal) minimum tours are computed for each cluster. The tours are concatenated using the midpoints yielding a solution for the original TSP. The method is tested on random TSPs independent, identically distributed points in the unit square) up to 10000 points as well as on a popular benchmark problem (att532 - coordinates of 532 American cities). Our solutions are 8-13% longer than the optimal ones. We also present an optimization algorithm for the partitioning to improve our solutions. This algorithm further reduces the solution errors (by several percent using 1000 iteration steps). The numerical experiments demonstrate the viability of the approach.

1 Introduction

The Traveling Salesman Problem (TSP) has generated a tremendous amount of research. It is an easy to state, yet hard to solve combinatorial problem: given n points, find the minimum length tour connecting all points that starts and ends at the same point. The TSP is an NP-hard problem [1]. Thorough studies of the TSP are given by Laporte [2] and Applegate et. al [3], the latter includes vast amount of computational results. Another overview of exact and approximate methods is given by Matai et al. [4], as well as examples of possible applications including vehicle routing and drilling of printed circuit boards.

The most popular exact methods for solving the TSP are the branch-and-bound methods [5, 6] and the branch-and-cut method [3]. The main drawback of these (and other) exact methods is the extensive amount of computer time they require to find the optimal solution of large TSP instances (10000 points or more). Approximate methods were developed to provide high quality solutions [7] quickly. These heuristics are either tour construction or tour improvement type methods. Tour construction methods gradually build up the TSP solution. Approximate methods which rely on the use of spacefilling curves [8, 9, 10] and partitioning algorithms [11, 12, 13] are tour construction methods. Tour improvement methods generate an initial tour and try to alter this tour to become shorter. Tour improvement using genetic algorithm [14, 15, 16, 17] is a popular approximate method. Another tour improvement type method is the 2-opt method [18, 19] which reorders the TSP solution to reduce tour length. It takes a section from the tour and adds it in reverse order to form a new tour as shown in Figure 1.

∗dyadic@kalmarnagy.com †bak@ara.bme.hu

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(a)

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Figure 1: The part of the solution (a) before and (b) after the application of the 2-opt method.

In this example there are crossing edges in the original solution (Figure 1(a)). Points 4 and 5 are swapped (added in reverse order to the original solution), thus the crossing edges are removed and the solution becomes shorter (Figure 1(b)). The 2-opt method is often incorporated into other heuristics for local optimization. The generalization of 2-opt leads to the very fast Lin-Kernighan heuristic [20, 21, 22] which is currently considered one of the best approximate methods for the TSP. The major drawback of approximate methods is that they often get stuck at a local minimum. This can be prevented using a metaheuristic such as ant colony optimization [23] or simulated annealing [24, 25, 26]. An overview of these metaheuristics is given by Johnson et al. [27].

Focusing on partitioning algorithms, an influential piece of work by Karp [11] considers such algorithms for the approximate solution of large instances of the TSP. To reduce the complexity of the problem, partitioning algorithms subdivide the set of points into small groups, construct an optimum tour through each group, and then patch these tours together to form a closed tour through all the points. Figure 2 shows the basic idea.

(a) (b) (c) (d)

Figure 2: (a) A set of n points, (b) is first clustered into disjoint sets. (c) A subsolution is obtained for each cluster, (d) and these are concatenated.

First the starting set of points (depicted in Figure 2(a)) is “reduced”: the points are grouped into disjoint subsets (clusters) as shown in Figure 2(b). Good quality subsolutions are computed for each cluster of points (Figure 2(c)). A cluster of points is replaced with a representative point, and the subsolutions are concatenated (Figure 2(d)).

Yoshiyuki and Yoshiki [12] introduced a method in which the unit square was recursively subdivided into smaller squares (regular square tiling). After each division the points located in the same subsquare were substituted with the barycenter of that subsquare. For every subdivision an approximate solution was constructed for the barycenters, based on the solution in the previous step. The work of Ugajin [13] relies on Yoshiyuki’s method, including a moving-frame renormalization scheme. In this approach the tiles are generated based on an intensity function, which has peaks around clusters of points. Xiang et al. [28] proposed a method to partition the points into four segments (groups) based on the coordinates of the points. Four corner points were designated in advance to serve as endpoints for the tours computed in each segment. Another hierarchical approach is proposed by Houdayer et al. [15]. They combined renormalization with a genetic algorithm.

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In this work, we propose a new partition-and-cluster approach. The set of points is partitioned with dyadic tilings [29, 30], yielding clusters of points. A TSP solution is calculated for the barycenters of the clusters. A tour with different endpoints (subsolution) is calculated for the points of each cluster. These tours are concatenated based on the TSP solution of the barycenters, yielding the global solution. The global solution strongly depends on the tiling, but this dependence is unknown a priori. A large number of TSPs consisting of n independent, identically distributed points in the unit square is solved using random dyadic tilings to investigate the solution quality.

To improve the efficacy of the algorithm we introduce the Genetic Algorithm Enhanced Hierarchical Solution (GAEHS) method which borrows elements of the classical genetic algorithm [31, 32, 33]. The genetic operators randomly choose and modify parts of the tiling. The modified tiling is kept if it is better than its predecessor.

This paper is structured as follows: in Section 2 we describe dyadic tiling, its connection with labeled complete binary trees and how the set of points is partitioned. In Section 3 we explain how a global solution of the TSP can be constructed in a hierarchical manner. In Section 4 results are presented for random TSPs and in Section 5 GAEHS is discussed and the improvements are demonstrated.

2 Tiles, trees and partitioning

Tiling (or tessellation) is fully covering an area with smaller, non-overlapping plane figures. The partitioning of the plane to find objects intersecting a specified area is a common approach [34, 35]. Tiling has been applied to a range of applications, including image or audio processing [36] and meshing complex geometry [37].

2.1 Dyadic tiling

A dyadic rectangle (referred to as tile) in the unit square is defined as

T (a, b, s, t) = [a 2−s, (a+ 1) 2−s]× [b 2−t, (b+ 1) 2−t], (1)

where the 4-tuple (a, b, s, t) of nonnegative integers satisfy

0 ≤ a < 2 s, 0 ≤ b < 2 t. (2)

The order of a tile is l = t + s. The unit square itself is T (0, 0, 0, 0), the unique 0th order tile. An lth order dyadic tiling partitions the unit square into 2l tiles of area 2−l [29, 30]. There is a correspondence between dyadic tilings and labeled complete binary trees (HV -trees) [30, 34]. Figure 3 illustrates such a correspondence.

(a) (b) (c)

Figure 3: (a) A 2nd order dyadic tiling, (b) illustration of the subdivision process, and (c) the corresponding HV -tree.

Figure 3(b) illustrates the “cutting” process. The unit square is depicted on the top. The labels H and V indicate whether the tile above them was halved horizontally or vertically. The loose edges at the bottom correspond to the tiles shown next to them. These tiles constitute the tiling shown in Figure 3(a). If the

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loose edges and the tiles are removed from this figure, we get a so-called HV -tree as shown in Figure 3(c). The HV -tree uniquely defines the tiling. The height (or order) of the HV -tree equals to the order of the dyadic tiling. For a given HV -tree the 4-tuple (a, b, s, t) can be calculated for each tile. If the 4-tuple of an lth order tile is (a, b, s, t), the 4-tuple of its left (L) and right (R) l+ 1th order children tiles are calculated as

HL(a, b, s, t) = (a, 2b, s, t+ 1) HR(a, b, s, t) = (a, 2b+ 1, s, t+ 1) VL(a, b, s, t) = (2a, b, s+ 1, t)

VR(a, b, s, t) = (2a+ 1, b, s+ 1, t),

(3)

where HL, HR, VL, VR are R4 7→ R4 affine transformations. Although every HV -tree uniquely defines the c