New approximation algorithms for graph coloring

The problem of coloring a graph with the minimum number of colorsis well known to be NP-hard, even restricted to<?Pub Fmt italic>k<?Pub Fmt /italic>-colorable graphs for constant<?Pub Fmt italic>k<?Pub Fmt /italic> ≥ 3. This paper explores theapproximation problem of coloring<?Pub Fmt italic>k<?Pub Fmt /italic>-colorable graphs with as fewadditional colors as possible in polynomial time, with special focus onthe case of <?Pub Fmt italic>k<?Pub Fmt /italic> = 3. The previous best upper bound on the number of colors needed forcoloring 3-colorable <?Pub Fmt italic>n<?Pub Fmt /italic>-vertex graphsin polynomial time was <inline-equation><f><it>o<fen lp="par"><rad><rcd>n/<rad><rcd><rm>log<it>n</it></rm></rcd></rad></rcd></rad><rp post="par"></fen></it></f></inline-equation> colors by Berger and Rompel, improving a bound of<inline-equation><f><it>o<fen lp="par"><rad><rcd>n</rcd></rad><rp post="par"></fen></it></f></inline-equation> colors by Wigderson. This paper presents an algorithmto color any 3-colorable graph with <inline-equation><f><it>o<fen lp="par">n<sup>3/8<inf><rm>polylog<fen lp="par"><it>n</it><rp post="par"></fen></rm></inf></sup><rp post="par"></fen></it></f></inline-equation> <?Pub Caret1>colors, thus breaking an“<?Pub Fmt italic>O((n<supscrpt>1/2-&ogr;(1)</supscrpt>)<?Pub Fmt /italic>barrier”. The algorithm given here is based on examiningsecond-order neighborhoods of vertices, rather than just immediateneighborhoods of vertices as in previous approaches. We extend ourresults to improve the worst-case bounds for coloring<?Pub Fmt italic>k<?Pub Fmt /italic>-colorable graphs for constant<?Pub Fmt italic>k<?Pub Fmt /italic> > 3 as well.

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