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Finding Hidden Cliques in Linear Time with High Probability

Published online by Cambridge University Press:  14 November 2013

YAEL DEKEL ,
ORI GUREL-GUREVICH  and
YUVAL PERES
YAEL DEKEL
Affiliation:
The Selim and Rachel Benin School of Computer Science and Engineering, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 91904, Israel (e-mail: yaelvin@cs.huji.ac.il)
ORI GUREL-GUREVICH
Affiliation:
Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2Canada (e-mail: origurel@math.ucb.ca)
YUVAL PERES
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: peres@microsoft.com)
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Abstract

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

Keywords

Primary 05C80Secondary 05C8568W20
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 29 - 49
Copyright
Copyright © Cambridge University Press 2013 

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