Clique Is Hard on Average for Regular Resolution

17 Dec 2020 · Albert Atserias, Ilario Bonacina, Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, Alexander Razborov ·

We prove that for $k \ll \sqrt[4]{n}$ regular resolution requires length $n^{\Omega(k)}$ to establish that an Erd\H{o}s-R\'enyi graph with appropriately chosen edge density does not contain a $k$-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional $n^{\Omega(k)}$ lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

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Clique Is Hard on Average for Regular Resolution

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