1-skeletons of the spanning tree problems with additional constraints

TL;DR

This paper investigates the polyhedral and combinatorial properties of two constrained spanning tree problems, revealing NP-completeness in their 1-skeleton adjacency problems and fundamental differences from classical minimum spanning trees.

Contribution

It analyzes the 1-skeletons of the polyhedra for two constrained spanning tree problems, establishing NP-completeness of adjacency and highlighting key geometric differences from classical MST.

Findings

Determining vertex adjacency in the 1-skeleton is NP-complete for both problems.

Superpolynomial lower bounds on clique numbers suggest high complexity for related algorithms.

The problems exhibit fundamental geometric differences from classical minimum spanning trees.

Abstract

We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less or equal a given value. In the second problem, an additional constraint is the assumption that the degree of all vertices of the spanning tree does not exceed a given value. The decision versions of both problems are NP-complete. We consider the polytopes of these problems and their 1-skeletons. We prove that in both cases it is a NP-complete problem to determine whether the vertices of 1-skeleton are adjacent. Although it is possible to obtain a superpolynomial lower bounds on the clique numbers of these graphs. These values characterize the time complexity in a broad class of algorithms based on linear comparisons. The results indicate a fundamental difference in combinatorial and geometric properties between the considered problems and the classical minimum spanning tree problem.