NDLI: Enumeration of isolated cliques and pseudo-cliques

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Enumeration of isolated cliques and pseudo-cliques

Content Provider	ACM Digital Library
Author	Ito, Hiro Iwama, Kazuo
Copyright Year	2009
Abstract	In this article, we consider $\textit{isolated}$ cliques and $\textit{isolated}$ dense subgraphs. For a given graph $\textit{G},$ a vertex subset $\textit{S}$ of size $\textit{k}$ (and also its induced subgraph $\textit{G}(\textit{S}))$ is said to be $\textit{c}-isolated$ if $\textit{G}(\textit{S})$ is connected to its outside via less than $\textit{ck}$ edges. The number $\textit{c}$ is sometimes called the isolation factor. The subgraph appears more isolated if the isolation factor is smaller. The main result in this work shows that for a fixed constant $\textit{c},$ we can enumerate all $\textit{c}-isolated$ maximal cliques (including a maximum one, if any) in linear time. In more detail, we show that, for a given graph $\textit{G}$ of $\textit{n}$ vertices and $\textit{m}$ edges, and a positive real number $\textit{c},$ all $\textit{c}-isolated$ maximal cliques can be enumerated in time $\textit{O}($ $c^{4}$ $2^{2c}m).$ From this, we can see that: (1) if $\textit{c}$ is a constant, all $\textit{c}-isolated$ maximal cliques can be enumerated in linear time, and (2) if $\textit{c}$ &equlas; $\textit{O}(log$ $\textit{n}),$ all $\textit{c}-isolated$ maximal cliques can be enumerated in polynomial time. Moreover, we show that these bounds are tight. That is, if $\textit{f}(\textit{n})$ is an increasing function not bounded by any constant, then there is a graph of $\textit{n}$ vertices and $\textit{m}$ edges for which the number of $\textit{f}(\textit{n})-isolated$ maximal cliques is superlinear in $\textit{n}$ + $\textit{m}.$ Furthermore, if $\textit{f}(\textit{n})$ = ω(log $\textit{n}),$ there is a graph of $\textit{n}$ vertices and $\textit{m}$ edges for which the number of $\textit{f}(\textit{n})-isolated$ maximal cliques is superpolynomial in $\textit{n}$ + $\textit{m}.$ We next introduce the idea of pseudo-cliques. A $\textit{pseudo-clique}$ having an average degree α and a minimum degree β, denoted by $\textit{PC}(α,β),$ is a set $\textit{V}′$ ⊆ $\textit{V}$ such that the subgraph induced by $\textit{V}′$ has an average degree of at least α and a minimum degree of at least β. This article investigates these, and obtains some cases that can be solved in polynomial time and some other cases that have a superpolynomial number of solutions. Especially, we show the following results, where $\textit{k}$ is the number of vertices of the isolated pseudo-cliques: (1) For any ϵ > 0 there is a graph of $\textit{n}$ vertices for which the number of 1-isolated $\textit{PC}(\textit{k}$ ™ (log $\textit{k})1$ + ϵ, $\textit{k}/(log$ $\textit{k})1$ + ϵ) is superpolynomial, and (2) there is a polynomial-time algorithm which enumerates all $\textit{c}-isolated$ $\textit{PC}(\textit{k}$ ™ log $\textit{k},$ $\textit{k}/log$ $\textit{k}),$ for any constant $\textit{c}.$
Starting Page	1
Ending Page	21
Page Count	21
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/1597036.1597044
Volume Number	5
Issue Number	4
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2009-11-06
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Clique Enumeration Isolation
Content Type	Text
Resource Type	Article
Subject	Mathematics

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