A combination of clustering algorithms with Ant Colony Optimization for large clustered Euclidean Travelling Salesman Problem TRUNG HOANG DINH, ABDULLAH AL MAMUN Department of Electrical and Computer Engineering National University of Singapore, NUS Address: 10 Kent Ridge Crescent, Singapore 117584 Abstract: The algorithm of Ant Colony System (ACS) has been found attractive for solving combinatorial optimization problem such as Travelling Salesman Problem (TSP). The run-time for this algorithm increases with increase in number of nodes. In this paper, we propose an efficient method that reduces the run-time for very large-scale Euclidean TSP problem and yet conserve the quality of solution for certain clustered cases. Moreover, the proposed method has a simple parallel implementation. The proposed method shows excellent performance both in run-time and quality of solution specially on large clustered instances. Effectiveness of the proposed method is underscored by applying them on two kinds of different benchmark problems. Key-Words: travelling salesman problem, ant colony optimization, clustering algorithms, combinatorial optimization. 1 Introduction The classic well-known NP-hard Travelling Sales- man Problem (TSP) has been used as a rich testing ground for most important algorithmic ideas dur- ing the past few decades. Interested readers may refer to Lawler et al. [8] for a fascinating history. In Euclidean TSP, nodes lie in ℜ 2 (or more gener- ally, in ℜ d for some d), and the distance is defined using Euclidean norm. Moreover, similar to TSP, it belongs to the NP-hard class [9, 7]. Many algorithms have already been developed to solve TSP. They are categorized temporally follow- ing one of two strategies by mathematical exaction or approximation of resulted solutions. Algorithms such as branch and bound, branch and cut giving the solutions whose optimality is mathematically proven, may require a large amount of run-time for large instances. The other approach is to de- sign approximate algorithms whose performances are often evaluated based on both their run-time and quality of solutions. There have been many algorithms developed following such a way . One approach of designing approximation algo- rithms is based on meta-heuristics such as genetic algorithm (GA), simulated annealing (SA), and ant algorithms. Most ant algorithms which have been successfully applied to many combinatorial optimization problems [3] follows a general scheme which is called Ant Colony Optimization - ACO (see [2]). Ant System (AS) [5], AS’s variants eg. ant-Q [6], rank ant [1]), ACS [4] are some exam- ples of the ant algorithms applied to TSP problem. We chose ACS due to the fact that it is one of the most successful ant algorithms applied to TSP, outperformed GA, SA, and also meets conditions in the theorem about reducing run-time which will be discussed later. Our method of solving Euclidean TSP problem of very large instances is divided into three separated stages(in some scenes the first and the third stage can be combined together in one single stage). In the first stage, the original instance is partitioned into few clusters of smaller dimensions. Each clus- ter is considered as a sub-TSP, and in the second stage, ACS is applied to find the optimal solution of each sub-TSP. We also design an algorithm to combine the sub-tours found in the second stage in order to get a final solution of the original TSP instance. For the same settings, the run-time of 1