Spanning and Splitting: Integer Semidefinite Programming for the Quadratic Minimum Spanning Tree Problem

TL;DR

This paper introduces a novel semidefinite programming approach for the quadratic minimum spanning tree problem, providing improved bounds and computational efficiency over existing methods, especially for larger graphs.

Contribution

It formulates the QMSTP as a mixed-integer semidefinite program, derives a relaxation, and develops a splitting method to compute bounds more effectively.

Findings

Significantly improved bounds over existing methods.

Efficient computation for graphs with more than 30 vertices.

Demonstrates the effectiveness of semidefinite programming in combinatorial optimization.

Abstract

In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We present a formulation of the QMSTP as a mixed-integer semidefinite program exploiting the algebraic connectivity of a graph. Based on this formulation, we derive a doubly nonnegative relaxation for the QMSTP and investigate classes of valid inequalities to strengthen the relaxation using the Chv\'atal-Gomory procedure for mixed-integer conic programming. Solving the resulting relaxations is out of reach for off-the-shelf software. We therefore develop and implement a version of the Peaceman-Rachford splitting method that allows to compute the new bounds for graphs from the literature. The computational results demonstrate that our bounds significantly improve over existing bounds from the literature in both quality and computation time, in particular for graphs with more than 30 vertices. This work is further evidence that semidefinite programming is a valuable tool to obtain high-quality bounds for problems in combinatorial optimization, in particular for those that can be modelled as a quadratic problem.