A New Approach Towards Graph Coloring

Consider the problem of deciding whether a graph is 3-colorable. This problem is NP-complete. The optimization version of this problem is, given a 3-colorable graph G, color it using as few colors as possible. In the regime of polynomial time approximation algorithms, this problem has been studied extensively [Wig82, Blu94, KMS98, ACC06, Chl07]. The current best known approximation factor is achieved by an algorithm due to Kawarabayashi and Thorup [iKT12] which uses at most n0.2038 colors to color a 3-colorable graph. Surprisingly, very few lower bounds are known for the approximability of the chromatic number of a graph. Dinur, Mossel and Regev [DMR06] showed that it’s hard to color a 3-colorable graph with O(1) colors based on a variant of Unique Games Conjecture. Feige, Langberg and Schechtman [FLS02] showed that the simple SDP relaxation has an integrality gap of at least n0.157. In a recent work, Barak, Raghavendra and Steurer [BRS11] and Guruswami and Sinop [GS11] showed a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. In another recent work [BBH12], Barak et. al. gave a natural polynomial time algorithm based on the ‘Sum of Squares’ SDP hierarchy that solves the previously proposed hard instances for Unique Games [RS09, KS09, KPS10, BGH11]. Along these line, we propose to study a new approach towards sub-exponential time algorithms with improved approximation guarantees for combinatorial problems like graph coloring, vertex cover, etc. Barak et. al.’s algorithm [BRS11] can be viewed as a divide-and-conquer approach. Consider an instance I of MAX 2-CSP. The constraint graph G of I is first partitioned into sets with small ε-threshold rank 1 such that only O(ε) fraction of the edges are cut. This step is same as the preprocession step used in Arora, Barak and Steurer’s sub-exponential time algorithm for unique games [ABS10]. Barak et. al. show that graphs with low threshold rank are ‘easy’. More formally, they show that O(rank1−ε(G) levels of the Lasserre hierarchy is sufficient to recover an integral solution from the SDP’s vector solution whose cost2 is at most O(ε) lesser than that of the SDP solution.