NDLI: The Monotone Complexity of k-clique on Random Graphs

Content Provider	IEEE Xplore Digital Library
Author	Rossman, B.
Copyright Year	2010
Abstract	It is widely suspected that Erd\H{o}s-R\'enyi 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 $O(n^{k/4})$ solves the $k$-clique problem with high probability on $\ER(n,p)$ 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 {\em quasi-sunflowers}, a new relaxation of sunflowers in which petals may overlap slightly on average. A ``quasi-sunflower lemma'' (\`a la the Erd\H{o}s-Rado sunflower lemma) leads to our novel lower bounds within Razborov's method of approximations.
Starting Page	193
Ending Page	201
File Size	496375
Page Count	9
File Format	PDF
ISBN	9781424485253
ISSN	02725428
DOI	10.1109/FOCS.2010.26
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2010-10-23
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Complexity theory Approximation methods Lattices Logic gates Digital TV Approximation algorithms Additives quasi-sunflowers clique monotone circuits average-case complexity
Content Type	Text
Resource Type	Article

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