Finding hidden cliques in linear time with high probability

Yael Dekel, Ori Gurel-Gurevich, Yuval Peres

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

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 1/2. This random graph model is denoted G(n,1/2,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√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(n 2). 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(n 2) (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.

Original languageAmerican English
Pages (from-to)29-49
Number of pages21
JournalCombinatorics Probability and Computing
Volume23
Issue number1
DOIs
StatePublished - Jan 2014
Externally publishedYes

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