Affiliation:
1. Massachusetts Institute of Technology, Cambridge
Abstract
The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to
k
-colorable graphs for constant
k
≥ 3. This paper explores the approximation problem of coloring
k
-colorable graphs with as few additional colors as possible in polynomial time, with special focus on the case of
k
= 3.
The previous best upper bound on the number of colors needed for coloring 3-colorable
n
-vertex graphs in polynomial time was
O(√n / √log n
colors by Berger and Rompel, improving a bound of
O(√n)
colors by Wigderson. This paper presents an algorithm to color any 3-colorable graph with
O(n
3/8
polylog(
n
)) colors, thus breaking an “
O((n
1/2-o(1)
)
barrier”. The algorithm given here is based on examining second-order neighborhoods of vertices, rather than just immediate neighborhoods of vertices as in previous approaches. We extend our results to improve the worst-case bounds for coloring
k
-colorable graphs for constant
k
> 3 as well.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
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