A characterization of linearizable instances of the quadratic minimum spanning tree problem

TL;DR

This paper characterizes specific instances of the quadratic minimum spanning tree problem that can be efficiently solved as linear problems, providing verifiable criteria for various graph classes.

Contribution

It offers a characterization of quadratic minimum spanning tree instances reducible to linear problems, with efficient verification methods for certain graph types.

Findings

Characterization of quadratic MST instances solvable as linear MSTs.

Verification algorithms with $O(|E|^2)$ and $O(|E|)$ complexity.

Identification of open problems related to the problem class.

Abstract

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph $G=(V,E)$ that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in $O(|E|^2)$ time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in $O(|E|)$ time. Related open problems are also indicated.