The Quadratic Minimum Spanning Tree Problem and its Variations

TL;DR

This paper explores the quadratic minimum spanning tree problem and its variations, analyzing their complexity, identifying solvable cases, and providing new insights into their structure and special instances.

Contribution

It introduces new polynomially solvable cases, characterizes NP-hard instances, and offers formulas and characterizations related to these spanning tree problems.

Findings

Identified new polynomially solvable cases

Found NP-hard instances on simple graphs

Provided recursive formulas and matroid characterizations

Abstract

The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict pair constraints are useful in modeling various real life applications. All these problems are known to be NP-hard. In this paper, we investigate these problems to obtain additional insights into the structure of the problems and to identify possible demarcation between easy and hard special cases. New polynomially solvable cases have been identified, as well as NP-hard instances on very simple graphs. As a byproduct, we have a recursive formula for counting the number of spanning trees on a $(k,n)$-accordion and a characterization of matroids in the context of a quadratic objective function.