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- W2952489188 abstract "We study the coloring problem: Given a graph G, decide whether $c(G) leq q$ or $c(G) ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 le q < Q$. For $qge 4$, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our hardness result on a certain `fish shaped' variant of his conjecture. We also prove that the problem almost coloring is hard for any constant $eps>0$, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a $eps$ fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least $eps$. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05]." @default.
- W2952489188 created "2019-06-27" @default.
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- W2952489188 date "2005-04-14" @default.
- W2952489188 modified "2023-09-25" @default.
- W2952489188 title "Conditional Hardness for Approximate Coloring" @default.
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