A Steady-State Genetic Algorithm for the Tree t-Spanner Problem

Abstract

A tree t-spanner, in a given connected graph, is a spanning tree T in which the distance between every two vertices in T is at most t times their shortest distance in the graph, where the parameter t is called stretch factor of T. This paper studies the tree t-spanner problem on a given connected, undirected and edge-weighted graph G that seeks to find a spanning tree in G whose stretch factor is minimum among all tree spanners in G. This problem is \(\mathcal {NP}\)-Hard for any fixed \(t > 1\). In the domain of metaheuristic techniques, only genetic algorithm (GA) based on generational population model is proposed in the literature. This paper presents a steady-state genetic algorithm (SSGA) for this problem, which is quite different from the existing GA in the literature, not only in the population management strategy, but also in genetic operators. Genetic operators in SSGA use problem-specific knowledge for generating offspring, making SSGA highly effective in finding high-quality solutions in comparison to the existing GA. On a set of randomly generated instances, computational results justify the effectiveness of SSGA over the existing GA.