Strong bounds and exact solutions to the minimum broadcast time problem

TL;DR

This paper introduces a new integer programming model and bounding techniques for the NP-hard minimum broadcast time problem, demonstrating improved efficiency and solution quality in dense graphs.

Contribution

It presents a compact integer programming formulation and effective bounds, advancing the state-of-the-art in solving the minimum broadcast time problem.

Findings

Bounds collapse in dense graphs, indicating optimal solutions.

The new model outperforms previous models in efficiency.

Strong bounds help close the solution gap in many instances.

Abstract

Given a graph and a subset of its nodes, referred to as source nodes, the minimum broadcast problem asks for the minimum number of steps in which a signal can be transmitted from the sources to all other nodes in the graph. In each step, the sources and the nodes that already have received the signal can forward it to at most one of their neighbour nodes. The problem has previously been proved to be NP-hard. In the current work, we develop a compact integer programming model for the problem. We also devise procedures for computing lower bounds on the minimum number of steps required, along with methods for constructing near-optimal solutions. Computational experiments demonstrate that in a wide range of instances, in particular instances with sufficiently dense graphs, the lower and upper bounds under study collapse. In instances where this is not the case, the integer programming model proves strong capabilities in closing the remaining gap, and proves to be considerably more efficient than previously studied models.