Improving EAX with Restricted 2-opt Chen-hsiung Chan frankch@life.nthu.edu.tw Sheng-An Lee shengan@lis.idv.tw Cheng-Yan Kao cykao@csie.ntu.edu.tw Department of Computer Science and Information Engineering National Taiwan University No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan +886-2-2362-5336 ext. 401 Huai-Kuang Tsai d7526010@csie.ntu.edu.tw Genomic Research Center Academia Sinica No. 128, Sec. 2, Academia Road, Nankang, Taipei, 115, Taiwan ABSTRACT Edge Assembly Crossover (EAX) is by far the most successful crossover operator in solving the traveling salesman problem (TSP) with Genetic Algorithms (GAs). Various improvements have been proposed for EAX in GA. However, some of the improvements have to make compromises between performance and solution quality. In this work, we have combined several improvements proposed in the past, including heterogeneous pair selection (HpS), iterative child generation (ICG), and 2-opt. We also incorporate 2-opt into EAX, and restricted the 2-opt local searches to sub-tours in the intermediates generated by EAX. Our proposed method can improve the performance of EAX with decreased number of generations, error rates, and computation time. The applications of conventional 2-opt and our restricted 2- opt concurrently have additive effect on the performance gain, and this performance improvement is more obvious in larger problems. The proposed method also enhanced the solution quality of EAX. The significances of the restricted 2-opt and the conventional 2-opt in EAX were analyzed and discussed. Categories and Subject Descriptors Genetic Algorithm. General Terms Algorithms. Keywords Combinatorial optimization, Local Search, Genetic Algorithms, Traveling Salesman Problem (TSP), Edge Assembly Crossover (EAX), Restricted 2-opt. 1. INTRODUCTION The traveling salesman problem (TSP) is a well-known NP-hard optimization problem. The problem designates n vertices as n cities, and tries to find the shortest round tour visiting each city exactly once. Many problems in various fields can be formulated as TSP. For example, scheduling [1], physical mapping [2], and even protein folding [3] can be treated as TSP. Genetic algorithms (GAs) have been widely used in tackling TSP. Many works have tried to obtain the optimum tour lengths of TSPs and to minimize the computing costs with GAs. Various operations have to be considered when using GAs to solve TSP. These include crossover, mutation, and selection. Selection in GA including selection of parent pairs and selection for survival. In most GAs, the selection for survival process simply picks the tour with shortest tour length; but more complex selection schemes are also used. The crossover operators are more challenging, because simply swap segments in two tours will not produce valid solutions to TSP. Among the numerous crossover operators, edge-based operators, i.e. EX [4], EXX [5], and EAX [6], are of particular interest. Many researches have found that EAX performs well, and a number of improvements have been proposed. Edge-based operators may lead to diversity loss of the population. That is, the children will tend to become more similar to one of the parents. Mutation operations in GAs are intended to introduce diversity into the population. A neighbor-join (NJ) operator [7] has been shown to perform well, and can enhance the performance of GA with EAX operator. In this work, we proposed a new operator which incorporate mutation operations into EAX. The 2-opt local search has been applied to intermediates of tours generated in EAX. Unlike other improvements which either apply mutation operator to generated tours, or use 2-opt to generate initial population, our proposed method restricted the positions of 2-opt to sub-tours formed during EAX operation. The restricted 2-opt operator can increase the diversity within solutions generated by EAX. The edge replacements in EAX algorithm tend to generate children similar to one of the parents. If these children replace the dissimilar parent, the remaining individuals will become more similar, and cause the diversity loss in the population. Selection scheme has been proposed by Nagata [8] to consider diversity loss, but the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO’05, June 25-29, 2005, Washington, DC, USA. Copyright 2005 ACM 1-59593-010-8/05/0006…$5.00. 1471