An Improved Ant-Based Algorithm for Minimum Degree Spanning Tree Problems

Abstract

A spanning tree of a connected graph is a sub graph, with least number of edges that still spans. The problem of finding degree constraint spanning tree is known to be NP-hard. In this paper we discuss an Ant-Based algorithm for finding minimum degree spanning trees and give improvement of the algorithm. We also show comparisons among the three algorithms and find the best improved Ant-Based algorithm. Extensive experimental results show that our improved algorithm performs very well against other algorithms on a set of 50 problem instances. Keywords - Ant algorithm, Graph algorithms, Heuristic methods, minimum degree spanning tree, I. INTRODUCTION This paper describes Ant-Based algorithms for minimum degree spanning tree (AB-MDST) of unweighted connected graph. This is an interesting, real-world problem that seems well suited to an ant algorithm approach. The AB-MDST problem entails finding a spanning tree such that the maximum degree of a vertex in the tree becomes minimum (2). This concept is useful in the design of telecommunication networks, design of networks for computer communication, design of integrated circuits, energy networks, transportation, logistics, and sewage networks (3). For instance, switches in an actual communication network will have limited number of connections available. Transportation systems must place a limit on the number of roads meeting in one place. The problem of finding degree constraint spanning tree is NP-hard (9). Therefore, heuristics are often used to find good solutions in a reasonable amount of time (10). We have used one type of heuristic called Ant Colony Optimization (ACO) (10). Here, artificial ants move based on local information and pheromone levels. Our algorithm uses cumulative pheromone levels to determine candidate set of edges from which minimum degree spanning trees are built (7). In this paper, we compare 3 algorithms - AB-MDST without local search and without degree constraint, AB-MDST with local search but without degree constraint and the last one AB- MDST with local search and with degree constraint. Extensive experimental results show that AB-MDST with local search and with degree constraint performs very well against other algorithms. The rest of the paper is organized as follow. In section 2 Our Ant-Based algorithm and two improved versions of that algorithm are described. Section 3 compares the performances of the three algorithms. The conclusion is given in section 4.