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Very Large Cliques are Easy to Detect

Authors: Alexander E. Andreev and Stasys Jukna

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

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.

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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)

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@InProceedings{andreev_et_al:DagSemProc.06111.22,
  author =	{Andreev, Alexander E. and Jukna, Stasys},
  title =	{{Very Large Cliques are Easy to Detect}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.22},
  URN =		{urn:nbn:de:0030-drops-6092},
  doi =		{10.4230/DagSemProc.06111.22},
  annote =	{Keywords: Clique function, monotone circuits, perfect hashing}
}

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DOI: 10.4230/DagSemProc.06111.19

The optimal sequence compression

Authors: Alexander E. Andreev

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Abstract

This paper presents the optimal compression for sequences with undefined values. Let we have $(N-m)$ undefined and $m$ defined positions in the boolean sequence $vv V$ of length $N$. The sequence code length can't be less then $m$ in general case, otherwise at least two sequences will have the same code. We present the coding algorithm which generates codes of almost $m$ length, i.e. almost equal to the lower bound. The paper presents the decoding circuit too. The circuit has low complexity which depends from the inverse density of defined values $D(vv V) = frac{N}{m}$. The decoding circuit includes RAM and random logic. It performs sequential decoding. The total RAM size is proportional to the $$logleft(D(vv V) ight) ,$$ the number of random logic cells is proportional to $$log logleft(D(vv V) ight) * left(log log logleft(D(vv V) ight) ight)^2 .$$ So the decoding circuit will be small enough even for the very low density sequences. The decoder complexity doesn't depend of the sequence length at all.

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Alexander E. Andreev. The optimal sequence compression. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{andreev:DagSemProc.06111.19,
  author =	{Andreev, Alexander E.},
  title =	{{The optimal sequence compression}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.19},
  URN =		{urn:nbn:de:0030-drops-6025},
  doi =		{10.4230/DagSemProc.06111.19},
  annote =	{Keywords: Compression, partial boolean function}
}

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Documents authored by Andreev, Alexander E.

Very Large Cliques are Easy to Detect

Abstract

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The optimal sequence compression

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