Abstract
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (kepsilon)^pw(G)poly(n) time algorithm for deciding if an nvertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between kcolorable graphs having at most s proper kcolorings and nonkcolorable graphs. We also show how to obtain a kcoloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous AlonTarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is kcolorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a nonzero value if the graph has few kcolorings. Yet every nonkcolorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.
BibTeX  Entry
@InProceedings{bjrklund:LIPIcs:2016:6035,
author = {Andreas Bj{\"o}rklund},
title = {{Coloring Graphs Having Few Colorings Over Path Decompositions}},
booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
pages = {13:113:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770118},
ISSN = {18688969},
year = {2016},
volume = {53},
editor = {Rasmus Pagh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6035},
URN = {urn:nbn:de:0030drops60353},
doi = {10.4230/LIPIcs.SWAT.2016.13},
annote = {Keywords: Graph vertex coloring, path decomposition, AlonTarsi theorem}
}
Keywords: 

Graph vertex coloring, path decomposition, AlonTarsi theorem 
Collection: 

15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016) 
Issue Date: 

2016 
Date of publication: 

22.06.2016 