Strategies for generating tree spanners: Algorithms, heuristics and optimal graph classes

Abstract

The t-admissibility problem aims to decide whether a graph G has a spanning tree T in which the distance between any two adjacent vertices of G is at most t. Regarding its optimization version, we want to determine the smallest t=σT(G) for which G is t-admissible, i.e., the stretch index of G. The t-admissibility problem is NP-complete for σT(G)≤t, t≥4. We develop strategies and implementations of two exact brute-force algorithms: sequential and parallel, in such a way that we propose parallel strategies for generating all spanning trees of a graph. We also propose two greedy heuristics to obtain solutions for the t-admissibility problem. We evaluate these implementations on random Barabási-Albert graphs, Erdös-Rényi graphs, Watts-Strogatz graphs and Bipartite graphs. Moreover, we determine graph classes for which the heuristics lead us to the stretch index value, such as split graphs, cographs and p-cycle power graphs.