Network conduciveness with application to the graph-coloring and independent-set optimization transitions

TL;DR

This paper introduces the concept of network conduciveness to analyze solution space transitions in graph coloring and independent set problems, revealing how network structure facilitates finding optimal solutions at critical points.

Contribution

It defines a new measure of network conduciveness and demonstrates its effectiveness in explaining solution transitions in NP-hard graph problems.

Findings

Network conduciveness increases towards optimal solutions at transition points.

Transitions correspond to increased network conduciveness for optimal solutions.

Network structure changes explain the ease of finding solutions at critical thresholds.

Abstract

We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another. We exemplify its use through an application to the two problems in combinatorial optimization that, given an undirected graph, ask that its so-called chromatic and independence numbers be found. Though NP-hard, when solved on sequences of expanding random graphs there appear marked transitions at which optimal solutions can be obtained substantially more easily than right before them. We demonstrate that these phenomena can be understood by resorting to the network that represents the solution space of the problems for each graph and examining its conduciveness between the non-optimal solutions and the optimal ones. At the said transitions, this network becomes strikingly more conducive in the direction of the optimal solutions than it was just before them, while at the same time becoming less conducive in the opposite direction. We believe that, besides becoming useful also in other areas in which network theory has a role to play, network conduciveness may become instrumental in helping clarify further issues related to NP-hardness that remain poorly understood.