Degree-Constrained Minimum Spanning Tree Problem in Stochastic Graph

Abstract

A degree-constrained minimum spanning tree (DCMST) problem is an NP-hard combinatorial optimization problem in graph theory seeking the minimum cost spanning tree with the additional constraint on the vertex degree. Several different approaches have been proposed in the literature to solve this problem using a deterministic graph. However, to the best of the author's knowlege, no work has been performed on solving the problem using stochastic edgeweighted graphs. In this article, a learning automata-based algorithm is proposed to find a near optimal solution of the DCMST problem using a stochastic graph, where the cost associated with the graph edge is a random variable with a priori unknown probability distribution. The convergence of the proposed algorithm to the optimal solution is theoretically proved based on the Martingale theorem. To show the performance of the proposed algorithm, several simulation experiments are conducted on stochastic Euclidean graph instances. Numerical results are compared with those of the standard sampling method (SSM). The numerical results confirm the superiority of the proposed sampling technique over the SSM both in terms of the sampling rate and solution optimality.