The quadratic minimum spanning tree problem: lower bounds via extended formulations

TL;DR

This paper introduces a new sequence of polynomial-sized relaxations for the quadratic minimum spanning tree problem, leveraging extended formulations and cutting plane algorithms to achieve tighter bounds than existing methods.

Contribution

The authors develop a novel set of relaxations for QMSTP based on extended formulations, which are computationally efficient and outperform previous bounds.

Findings

New relaxations with polynomial constraints

Bounds outperform existing methods

Efficient solution via cutting plane algorithm

Abstract

The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of a graph such that the total interaction cost between pairs of edges in the tree is minimized. We first show that most of the bounding approaches for the QMSTP are closely related. Then, we exploit an extended formulation for the minimum spanning tree problem to derive a sequence of relaxations for the QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in the literature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solved by a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.