Scale-free spanning trees: complexity, bounds and algorithms

TL;DR

This paper introduces the problem of finding scale-free-like spanning trees in graphs, proves its computational hardness, explores structural properties, and proposes algorithms with experimental evaluation.

Contribution

It defines the $m$-SF and $s$-SF spanning tree problems, proves their NP-hardness, and develops ILP formulations and heuristics for practical solutions.

Findings

NP-hardness of $m$-SF and $s$-SF problems in various graph classes

Structural characterization of optimal solutions in split graphs

Effective heuristics and ILP formulations demonstrated experimentally

Abstract

We introduce and study the general problem of finding a most "scale-free-like" spanning tree of a connected graph. It is motivated by a particular problem in epidemiology, and may be useful in studies of various dynamical processes in networks. We employ two possible objective functions for this problem and introduce the corresponding algorithmic problems termed $m$-SF and $s$-SF Spanning Tree problems. We prove that those problems are APX- and NP-hard, respectively, even in the classes of cubic, bipartite and split graphs. We study the relations between scale-free spanning tree problems and the max-leaf spanning tree problem, which is the classical algorithmic problem closest to ours. For split graphs, we explicitly describe the structure of optimal spanning trees and graphs with extremal solutions. Finally, we propose two Integer Linear Programming formulations and two fast heuristics for the $s$-SF Spanning Tree problem, and experimentally assess their performance using simulated and real data.