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SIAM Journal on Computing
We study the AprxColoring(q,Q) problem: Given a graph G, decide whether Χ(G) ≤ q or Χ(G)≥Q. We present hardness results for this problem for any constants 3 ≤ q < Q. For q ≥ 4, our result is base on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775]. For q=3, we base our hardness result on a certain “⋉-shaped" variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz ["Noise stability of functions with low influences: Invariance and optimality," Ann. of Math. (2), to appear] and should have wider applicability.
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hardness of approximation, unique games, graph coloring
Dinur, I., Mossel, E., & Regev, O. (2009). Conditional Hardness for Approximate Coloring. SIAM Journal on Computing, 39 (3), 783-1218. http://dx.doi.org/10.1137/07068062X
Date Posted: 27 November 2017
This document has been peer reviewed.