Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T16:56:58.998Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
ORI GUREL-GUREVICH
Affiliation:
Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2Canada (e-mail: [email protected])
YUVAL PERES
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abramowitz, M. and Stegun, I. A., eds (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover.Google Scholar
[2]Alon, N., Andoni, A., Kaufman, T., Matulef, K., Rubinfeld, R. and Xie, N. (2007) Testing k-wise and almost k-wise independence. In STOC, ACM, pp. 496–505.Google Scholar
[3]Alon, N., Krivelevich, M. and Sudakov, B. (1998) Finding a large hidden clique in a random graph. Random Struct. Alg. 13 457–466.Google Scholar
[4]Arora, S. and Safra, S. (1998) Probabilistic checking of proofs: A new characterization of NP. J. Assoc. Comput. Mach. 45 70–122.Google Scholar
[5]Arora, S., Lund, C., Motwani, R., Sudan, M. and Szegedy, M. (1992) Proof verification and hardness of approximation problems. In FOCS, ACM, pp. 14–23.Google Scholar
[6]Azuma, K. (1967) Weighted sums of certain dependent random variables. Tohoku Math. J. 19 357–367.Google Scholar
[7]Ben-Dor, A., Shamir, R. and Yakhini, Z. (1999) Clustering gene expression patterns. J. Comput. Biol. 6 281–297.Google Scholar
[8]Ben Sasson, E., Bilu, Y. and Gutfreund, D. (2002) Finding a randomly planted assignment in a random 3CNF. Manuscript.Google Scholar
[9]Berry, A. C. (1941) The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 122–136.CrossRefGoogle Scholar
[10]Durrett, R. (2010) Probability: Theory and Examples, fourth edition, Cambridge University Press.Google Scholar
[11]Esseen, C. G. (1942) On the Liapunoff limit of error in the theory of probability. Arkiv för Matematik, Astronomi och Fysik A28 1–19.Google Scholar
[12]Feige, U. and Ron, D. (2010) Finding hidden cliques in linear time. In AOFA, DMTCS.Google Scholar
[13]Feige, U., Goldwasser, S., Lovász, L., Safra, S. and Szegedy, M. (1991) Approximating clique is almost NP-complete (preliminary version). In FOCS, IEEE Computer Society, pp. 2–12.Google Scholar
[14]Feige, U. and Krauthgamer, R. (2000) Finding and certifying a large hidden clique in a semirandom graph. Random Struct. Alg. 16 195–208.Google Scholar
[15]Feige, U. and Krauthgamer, R. (2003) The probable value of the Lovász–Schrijver relaxations for maximum independent set. SIAM J. Comput. 32 345–370.Google Scholar
[16]Frieze, A. M. and Kannan, R. (2008) A new approach to the planted clique problem. In FSTTCS, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 187–198.Google Scholar
[17]Grimmett, G. and McDiarmid, C. (1975) On colouring random graphs. Math. Proc. Cam. Phil. Soc. 77 313–324.Google Scholar
[18]Hazan, E. and Krauthgamer, R. (2009) How hard is it to approximate the best Nash equilibrium? In SODA, Society for Industrial and Applied Mathematics, pp. 720–727.Google Scholar
[19]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[20]Jerrum, M. (1992) Large cliques elude the metropolis process. Random Struct. Alg. 3 347–359.Google Scholar
[21]Juels, A. and Peinado, M. (2000) Hiding cliques for cryptographic security. Des. Codes Cryptography 20 269–280.Google Scholar
[22]Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations (Miller, R. E. and Thatcher, J. W., eds), Plenum, pp. 85–103.Google Scholar
[23]Krivelevich, M. and Vilenchik, D. (2006) Solving random satisfiable 3CNF formulas in expected polynomial time. In SODA, ACM, pp. 454–463.Google Scholar
[24]Kučera, L. (1991) A generalized encryption scheme based on random graphs. In Graph-Theoretic Concepts in Computer Science (Schmidt, G. and Berghammer, R., eds), Vol. 570 of Lecture Notes in Computer Science, Springer, pp. 180–186.Google Scholar
[25]Kučera, L. (1995) Expected complexity of graph partitioning problems. Discrete Applied Math. 57 193–212.Google Scholar
[26]McSherry, F. (2001) Spectral partitioning of random graphs. In FOCS, IEEE Computer Society, pp. 529–537.Google Scholar