New Approximation Algorithms for Graph Coloring
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.