The monotone complexity of k-clique on random graphs


Conference Paper

It is widely suspected that Erdocombining double acute accents-Rényi random graphs are a source of hard instances for clique problems. Giving further evidence for this belief, we prove the first average-case hardness result for the k-clique problem on monotone circuits. Specifically, we show that no monotone circuit of size O(nk/4) solves the k-clique problem with high probability on G(n, p) for two sufficiently far-apart threshold functions p(n) (for instance n-2/(k-1) and 2n-2/(k-1)). Moreover, the exponent k/4 in this result is tight up to an additive constant. One technical contribution of this paper is the introduction of quasi-sunflowers, a new relaxation of sunflowers in which petals may overlap slightly on average. A "quasi-sunflower lemma" (à la the Erdocombining double acute accents-Rado sunflower lemma) leads to our novel lower bounds within Razborov's method of approximations. © 2010 IEEE.

Full Text

Duke Authors

Cited Authors

  • Rossman, B

Published Date

  • December 1, 2010

Published In

Start / End Page

  • 193 - 201

International Standard Serial Number (ISSN)

  • 0272-5428

International Standard Book Number 13 (ISBN-13)

  • 9780769542447

Digital Object Identifier (DOI)

  • 10.1109/FOCS.2010.26

Citation Source

  • Scopus