Guided local search and its application to the traveling salesman problem

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Abstract

The Traveling Salesman Problem (TSP) is one of the most famous problems in combinatorial optimization. In this paper, we are going to examine how the techniques of Guided Local Search (GLS) and Fast Local Search (FLS) can be applied to the problem. GLS sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems. GLS can be combined with the neighborhood reduction scheme of FLS which significantly speeds up the operations of the algorithm. The combination of GLS and FLS with TSP local search heuristics of different eiciency and effectiveness is studied in an effort to determine the dependence of GLS on the underlying local search heuristic used. Comparisons are made with some of the best TSP heuristic algorithms and general optimization techniques which demonstrate the advantages of GLS over alternative heuristic approaches suggested for the problem.

Introduction

The Traveling Salesman Problem (TSP) is one of the most famous combinatorial optimization problems. The problem is known to be NP-hard and over the years has been the testing ground for numerous techniques inspired from a variety of sources. Nowadays, TSP plays a very important role in the development, testing and demonstration of new optimization techniques. In this context, we are presenting the application to the TSP of a new metaheuristic approach called Guided Local Search (GLS) and its accompanying neighborhood reduction scheme called Fast Local Search (FLS).

GLS originally proposed by Voudouris and Tsang [47] is a general optimization technique suitable for a wide range of combinatorial optimization problems. Successful applications of the technique so far include practical problems such as Frequency Allocation [47], Workforce Scheduling [45] and Vehicle Routing [2,25] and also classic problems such as the TSP, Quadratic Assignment Problem (QAP) and Global Optimization [48]. In this paper, we present the technique to the wider Operations Research (OR) audience by explaining its application to the TSP, a widely known problem in the OR community.

GLS belongs to a class of techniques known as Meta-heuristics [37,38,40]. Prominent members of this class include Tabu Search [[12], [13], [14], [15], [16], [17], [18]], Simulated Annealing [1,9,26,28], GRASP [10], Genetic Algorithms [8,19,39], Scatter Search [13] and others. Meta-heuristics aim at enhancing the performance of heuristic methods in solving large and difficult combinatorial optimization problems.

In the case of GLS, the main focus is on the exploitation of problem and search-related information to effectively guide local search heuristics in the vast search spaces of NP-hard optimization problems. This is achieved by augmenting the objective function of the problem to be minimized with a set of penalty terms which are dynamically manipulated during the search process to steer the heuristic to be guided. Higher goals, such as the distribution of the search effort to the areas of the search space according to the promise of these areas to contain high quality solutions, can be expressed and pursued.

GLS is closely related to the Frequency-Based Memory approaches introduced in Tabu Search [14,18], extending these approaches to take into account the quality of structural parts of the solution and also react to feedback from the local optimization heuristic under guidance.

The paper is structured as follows. We first describe the basics of local search which is the foundation for most meta-heuristics. Following that we explain the different components of GLS and how it can be combined with the sister scheme of FLS particularly suited for speeding up the search of neighborhoods when GLS is used. The rest of the paper is devoted to the application of GLS and FLS to the famous TSP when these are combined with commonly used heuristics such as 2-Opt, 3-Opt and Lin–Kernighan. The benefits from using GLS and FLS with these heuristics are demonstrated and the dependence of GLS on them is investigated. Conclusions are drawn on the relation between GLS and the underlying local search procedures. Finally comparisons are conducted with other well-known general or TSP-specific metaheuristic techniques such as Simulated Annealing, Tabu Search, Iterated Lin–Kernighan and Genetic Algorithms. GLS is shown to perform equally well compared with state-of-the-art specialized methods while outperforming classic variants of well-known general optimization techniques. In all cases, publicly available TSP instances are used for which the optimal solutions are known so that the performance of algorithms can be measured with respect to approximating the optimal solutions.

Section snippets

Local search

Local Search, also referred to as Neighborhood Search or Hill Climbing, is the basis of many heuristic methods for combinatorial optimization problems. In isolation, it is a simple iterative method for inding good approximate solutions. The idea is that of trial and error. For the purposes of explaining local search, we will consider the following deinition of a combinatorial optimization problem.

A combinatorial optimization problem is defined by a pair (S, g), where S is the set of all

Guided local search

GLS has its root in a Neural Network architecture named GENET developed by Wang and Tsang [49]. GENET is applicable to a class of problems known as Constraint Satisfaction Problems [46] which are closely related to the class of SAT problems. GLS generalizes some of the elements present in the GENET architecture and applies them to the general class of combinatorial optimization problems. For more information on GENET and related techniques for CSP and SAT problems the reader can refer to Refs. [

Simulated annealing

Non-monotonic temperature reduction schemes used in Simulated Annealing (SA) also referred to as re-annealing or re-heating schemes are of interest in relation to the work presented in this paper. In these schemes, the temperature is decreased as well as increased in a attempt to remedy the problem that the annealing process eventually settles down failing to continuously explore good solutions. In a typical SA, good solutions are mainly visited during the mid and low parts of the cooling

The traveling salesman problem

In the previous sections, we examined the method of GLS and its generic framework. We are now going to examine the application of the method to the well-known Travelling Salesman Problem (TSP). There are many variations of the problem. In this work, we examine the classic symmetric TSP. The problem is defined by N cities and a symmetric distance matrix D = [dij] which gives the distance between any two cities i and j. The goal in TSP is to find a tour (i.e. closed path) which visits each city

Local search heuristics for the TSP

Local search for the TSP is synonymous with k-Opt moves. Using k-Opt moves, neighboring solutions can be obtained by deleting k edges from the current tour and reconnecting the resulting paths using k new edges. The k-Opt moves are the basis of the three most famous local search heuristics for the TSP, namely 2-Opt [6], 3-Opt [31] and Lin–Kernighan (LK) [32]. These heuristics deine neighborhood structures which can be searched by the different neighborhood search schemes described in Sections 2

Solution features and augmented cost function

The first step in the process of applying GLS to a problem is to ind a set of solution features that are accountable for part of the overall solution cost. For the TSP, a tour includes a number of edges and the solution cost (tour length) is given by the sum of the lengths of the edges in the tour (see Eq. (3)). Edges are ideal features for the TSP. First, they can be used to deine solution properties (a tour either includes an edge or not) and second, they carry a cost equal to the edge

Evaluation of GLS in the TSP

To investigate the behavior of GLS on the TSP, we conducted a series of experiments. The results presented in subsequent sections attempt to provide a comprehensive picture of the performance of GLS on the TSP. First, we examine the combination of GLS with 2-Opt, the simplest of the TSP heuristics. The beneits from using FLS instead of best improvement local search are clearly demonstrated, along with the ability of GLS to ind high quality solutions in small to medium size problems. These

Summary and conclusions

In this paper, we described the technique of GLS in detail and examined its application to the TSP. Eight combinations of GLS with commonly used TSP heuristics were described and evaluated on publicly available instances of the TSP. GLS with FLS-2Opt was found to be the best GLS variant for the TSP. The variant was compared and found to be superior to commonly used variants of general search methods such as SA and tabu search. Furthermore, we demonstrated that GLS with FLS-2Opt is highly

Acknowledgements

We would like to thank Nader Azarmi at BT Laboratories, without his encouragement and support this work may have never been completed. We would also like to thank the Department of Computer Science at the University of Essex for the excellent computer facilities without which it would have been impossible to perform the computational experiments reported in this paper. This research has been partially conducted in the framework of the GENET project funded by the EPSRC grant (GR/H75275).

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