New Tools for Graph Coloring

How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that lift and project based SDP relaxations could be used to design algorithms that use n colors for arbitrarily small > 0. We explore this possibility in this paper and find some cause for optimism. While the case of general graphs is till open, we can analyse the Lasserre relaxation for two interesting families of graphs. For graphs with low threshold rank (a class of graphs identified in the recent paper of Arora, Barak and Steurer on the unique games problem), Lasserre relaxations can be used to find an independent set of size Ω(n) (i.e., progress towards a coloring with O(log n) colors) in nO(D) time, where D is the threshold rank of the graph. This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre...

• Type: Journal
• Year: 2011
• Where: APPROX
• Authors: Sanjeev Arora, Rong Ge
• Keywords: Algorithms, APPROX 2011, Colorable Graphs, Family Of Graphs, Polynomial Time Algorithm