A Novel Ant Colony Optimization-Based Algorithm for the Optimal Communication Spanning Tree Problem

Abstract

Abstract-The optimal communication spanning tree (OCST) problem finds a spanning tree that connects all node satisfies their communication requirements for a minimum total cost. In this paper, we present a new method of finding optimal solution for OCST problem based on Ant Colony Optimization (ACO) to reduce search space mentioned above but still converge to a global good solution. Our algorithm take account into node biased encoding (NBE) scheme to find nearly optimal solution. The new algorithm can achieve a result that is better than known heuristic algorithms do, as verified by a set of public benchmark problem instances.Index Terms-Optimal communication spanning tree, node biased encoding, ant colony optimization. I. INTRODUCTIONThe problem of finding optimal communication spanning tree was introduced by Hu in 1974 [1]. Its NP-hard property was shown by Johnson et al [2]. The formal definition of this problem is below. Let The function f is defined as the product of two variables:The objective function is then:where  is the set of n n-2 label spanning tree of G [3]. We take the Palmer's 6-nodes instance from [4] as an example to clear out the definition.Example 1: The Palmer's 6-nodes instance.The distant matrix and requirement matrix are given below : 0 16661 18083 21561 21099 13461 16661 0 5658 9194 8797 10440 18083 5658 0 7230 6899 11340 21561 9194 7230 0 4300 13730 21099 8797 6899 4300 0 13130 13461 10440 11340 13730 13130 0 (a) The distant matrix 0 1 1 1 1 2 1 0 10 3 4 3 1 10 0 5 6 2 1 3 5 0 31 2 1 4 6 31 0 2 2 3 2 2 2 The cost of the tree T in Fig.2 For this instance, the optimal solution is given in Fig.2.b with the minimum cost is 693180.It is important to note that the OCST problem is far more different from the minimum spanning tree problem, which is already solved by several polynomial algorithms (such as Prim [5] and Kruskal). Moreover, there are several variations of this problem when the form of the distant matrix D or the requirement matrix R changes. More information can be found in [6]. II. RELATED WORKS AND OUR WORKSDue to its large application, there are many researched on OCST problem, from both exact and heuristic approaches. Authors in [1] defined the problem and discussed two simplest cases. The NP-hard property of the problem was showed [2] and Reshef showed that the problem is even a MAX SNP-hard one, which cannot be solved by a polynomial approximation algorithm unless P=NP [6].Therefore, heuristic optimization methods have been chosen by researchers to find optimal or nearly optimal solutions for the OCST problems. One of the first heuristic approaches was presented by Ahuja and Murty, which is closes to local search methods. Palmer, in his PhD thesis [8], proposed a genetic algorithm (GA) using the node and link biased encoding that will be clearly descript in section III.A. From the same approach, Li and Bouchebaba proposed another GA which works on a tree chromosome without intermediate encoding and decod...