An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the Hardness of 4-Coloring a 3-Colorable Graph
SIAM Journal on Discrete Mathematics
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Conditional Hardness for Approximate Coloring
SIAM Journal on Computing
Linear index coding via semidefinite programming
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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For 3 ≤ q Q we consider the ApproxColoring(q,Q) problem of deciding whether χ(G) ≤ q or &chi(G) ≥ Q for a given graph G. Hardness of this problem was shown in [7] for q = 3,4 and arbitrary large constant Q under variants of the Unique Games Conjecture [10]. We extend this result to values of Q that depend on the size of a given graph. The extension depends on the parameters of the conjectures we consider. Following the approach of [7], we find that a careful calculation of the parameters gives hardness of coloring a 4-colorable graph with lgc(lg(n)) colors for some constant c 0. By improving the analysis of the reduction we show that under related conjectures it is hard to color a 4-colorable graph with lgc(n) colors for some constant c 0. The main technical contribution of the paper is a variant of the Majority is Stablest Theorem, which says that among all balanced functions whose each coordinate has o(1) influence, the Majority function has the largest noise stability. We adapt the theorem for our applications to get a better dependency between the parameters required for the reduction.