Why almost all k-colorable graphs are easy

Amin Coja-Oghlan, Michael Krivelevich, Dan Vilenchik

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

Coloring a k-colorable graph using k colors (k ≥ 3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly en edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs are clustered in one cluster, and agree on all but a small, though constant, number of vertices. We also describe a polynomial time algorithm that finds a proper k-coloring for (1 -o(l))fraction of such random k-colorable graphs, thus asserting that most of them are "easy". This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more "regular" structure that denser graphs possess. Thus in some sense, our result rigorously supports this explanation.

Original languageEnglish
Title of host publicationSTACS 2007 - 24th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings
PublisherSpringer Verlag
Pages121-132
Number of pages12
ISBN (Print)9783540709176
DOIs
StatePublished - 1 Jan 2007
Externally publishedYes
Event24th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2007 - Aachen, Germany
Duration: 22 Feb 200724 Feb 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4393 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference24th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2007
Country/TerritoryGermany
CityAachen
Period22/02/0724/02/07

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