An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs

TL;DR

This paper introduces an integer programming approach with enhancements and a new lower bound for calculating the strong rainbow connection number of graphs, demonstrating practical effectiveness on large graphs.

Contribution

The paper presents the first computational methods for the strong rainbow connection problem, including an integer program, heuristics, and a novel lower bound.

Findings

Effective computation of src(G) for graphs up to 379 vertices

Introduction of a new lower bound for src(G)

Demonstration of the efficiency of iterative lower bound improvement

Abstract

We present an integer programming model to compute the strong rainbow connection number, $src(G)$, of any simple graph $G$. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound for $src(G)$ which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to $379$ vertices.