Volume 4, No. 11, Nov-Dec 2013 International Journal of Advanced Research in Computer Science RESEARCH PAPER Available Online at www.ijarcs.info © 2010, IJARCS All Rights Reserved 1 ISSN No. 0976-5697 Improvements on Heuristic Algorithms for Solving Traveling Salesman Problem Fidan Nuriyeva Institute of Cybernetics Azerbaijan National Academy of Sciences Baku, Azerbaijan nuriyevafidan@gmail.com Gözde Kızılateş Department of Mathematics Faculty of Science, Ege University Izmir, Turkey gozde.kizilates@gmail.com Murat Erşen Berberler Department of Computer Science Faculty of Science, Dokuz Eylul University Izmir, Turkey murat.berberler@deu.edu.tr Abstract: In this paper, four new heuristics are proposed in order to solve the traveling salesman problem. Comparisons are made between the results obtained from those heuristics. A new version of 2-opt and 3-opt methods are developed namely as 2-opt + 3-opt Shifting method. In addition, a new hybrid algorithm based on NN and Greedy algorithms is proposed. Computational experiments and comparisons are made on library problems for Hybrid, NN, and Greedy algorithms. Obtained results show the efficiency of the algorithms. Keywords: Traveling salesman problem; heuristic algorithms; hyper-heuristic algorithms; hybrid algorithms I. INTRODUCTION The traveling salesman problem (TSP) is a well-known and important combinatorial optimization problem [11]. The goal is to find the shortest (least expensive) tour that visits each city (node) in a given list exactly once and then returns to the starting city. In other words, TSP can be considered as a graph problem in which vertices represent cities and distances between cities are represented by edges. Formally, the TSP can be stated as follows: The distances between n cities are stored in a distance matrix D with elements ij d where , 1,..., ij n = and the diagonal elements ii d are zero. A tour can be represented by a cyclic permutation π of {1,2,..., } n where i π represents the city that follows city i on the tour. The traveling salesman problem is then the optimization problem of finding a permutation π that minimizes the length of the tour denoted by 1 () n i i d i π = ∑ . In this paper we shall concentrate on the symmetric TSP, in which the distances satisfy (, ) (,) dij d ji = for 1 , ij n ≤ ≤ . There are many variations of TSP: Symmetric TSP, Asymmetric TSP, The MAX TSP, The MIN TSP, TSP with multiple visits (TSPM), TSP with a closed tour, TSP with an open tour [3]. There are many variations of the problem. In this work, we examine the classic symmetric TSP. Solving TSP is an important part of many applications in different fields including vehicle routing, computer wiring, machine sequencing and scheduling, frequency assignment in communication networks as well as data analysis in psychology and clustering in biostatistics [12, 17]. For example, data analysis applications in psychology ranging from profile smoothing to finding an order in developmental data are presented by [5]. Clustering and ordering using TSP solvers are currently becoming popular in biostatistics [2, 13]. For example, [18] described an application for ordering genes and [9] used a TSP solver for clustering proteins. Given that the problem is NP-Hard, and hence the polynomial-time algorithms for finding optimal tours are unlikely to exist, much attention has been addressed to the question of efficient heuristic algorithms, fast algorithms that attempt only to find near-optimal tours. The rest of this paper is organized as follows. Section 2 describes some approaches for solving the TSP. Section 3 presents basic tour constructing algorithms such as NN and Greedy. Section 4 presents our new tour constructing proposed heuristics. Section 5 presents a new version of 2- opt and 3-opt algorithms that we have proposed. Section 6 presents other improved algorithms. Finally, section 7 concludes the paper. II. APPROACHES FOR SOLVING TSP Although definition of the TSP is easy, it belongs to NP- hard [6]. There are a number of algorithms used to find optimal tours, because of this problem is NP-hard, none are feasible for large instances since they all grow exponentially. That’s why heuristic algorithms are useful for this problem. The following approaches are developed for solving TSP. A. Exact Approaches: These approaches usually utilize the integer linear programming model of the TSP. “Branch & Bound” is one of the examples for this category [10]. One approach that comes to mind first is to try all possibilities. Other approach can be dynamic programming [4]. But these approaches are