TY - GEN

T1 - Why almost all k-colorable graphs are easy

AU - Coja-Oghlan, Amin

AU - Krivelevich, Michael

AU - Vilenchik, Dan

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38049136871&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-70918-3_11

DO - 10.1007/978-3-540-70918-3_11

M3 - Conference contribution

AN - SCOPUS:38049136871

SN - 9783540709176

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 121

EP - 132

BT - STACS 2007 - 24th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings

PB - Springer Verlag

T2 - 24th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2007

Y2 - 22 February 2007 through 24 February 2007

ER -