Improving the performance guarantee for approximate graph coloring
Journal of the ACM (JACM)
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
New approximation algorithms for graph coloring
Journal of the ACM (JACM)
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Conditional hardness for approximate coloring
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Linear degree extractors and the inapproximability of max clique and chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
A note on the /spl theta/ number of Lovasz and the generalized Delsarte bound
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Non-local analysis of sdp-based approximation algorithms
Non-local analysis of sdp-based approximation algorithms
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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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 SDP hierarchies could be used to design algorithms that use nε colors for arbitrarily small ε 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 Hierarchy for unique games. The algorithm can also be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserre lifting, which also seems reason for optimism. For distance transitive graphs of diameter Δ, we can show how to color them using O(log n) colors in n2O(Δ) time. This family is interesting because the family of graphs of diameter O(1/ε) is easily seen to be complete for coloring with nε colors. The distance-transitive property implies that the graph "looks" the same in all neighborhoods. The full version of this paper can be found at: http://www.cs.princeton.edu/~rongge/LasserreColoring.pdf .