Finding Hidden Cliques in Linear Time with High Probability

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 th...

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Bibliographic Details
Published in:Combinatorics, probability & computing Vol. 23; no. 1; pp. 29 - 49
Main Authors: DEKEL, YAEL, GUREL-GUREVICH, ORI, PERES, YUVAL
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.01.2014
Subjects:
Algorithms
Combinatorial analysis
Computation
Design engineering
Graphs
Polynomials
Running
Spectra
68W20
Primary 05C80
Secondary 05C85
ISSN:0963-5483, 1469-2163
Online Access:Get full text
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Summary: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.
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ISSN:0963-5483
1469-2163
DOI:10.1017/S096354831300045X