DagSemProc.06111.22.pdf
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Very Large Cliques are Easy to Detect
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Dagstuhl Seminar Proceedings, Volume 6111
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
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- Publication Date: 2006-11-20
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Alexander E. Andreev and Stasys Jukna. Very Large Cliques are Easy to Detect. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)
https://doi.org/10.4230/DagSemProc.06111.22
https://doi.org/10.4230/DagSemProc.06111.22
Abstract
It is known that, for every constant $kgeq 3$, the presence of a
$k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex
graph cannot be detected by a monotone boolean circuit using fewer
than $Omega((n/log n)^k)$ gates. We show that, for every constant
$k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be
detected by a monotone circuit using only $O(n^2log n)$ gates.
Moreover, if we allow unbounded fanin, then $O(log n)$ gates are
enough.
Keywords
- Clique function
- monotone circuits
- perfect hashing
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