Finding a large hidden clique in a random graph

We consider the following probabilistic model of a graph on n labeled vertices. . First choose a random graph Gn ,1 r 2 ,and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k ) cn 0.5 , for ˇ 0.5 . 0.5 any fixed c ) 0, thus improving the trivial case k ) cn log n . The algorithm is based on the spectral properties of the graph. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 457)466, 1998

[1]  Alan M. Frieze,et al.  Algorithmic theory of random graphs , 1997, Random Struct. Algorithms.

[2]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[3]  D. Welsh,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1994 .

[4]  Mark Jerrum,et al.  Large Cliques Elude the Metropolis Process , 1992, Random Struct. Algorithms.

[5]  M. A. Wolfe.  A first course in numerical analysis , 1972 .

[6]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  Ludek Kucera,et al.  Expected Complexity of Graph Partitioning Problems , 1995, Discret. Appl. Math..

[8]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[9]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[10]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[11]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[12]  Noga Alon,et al.  Approximating the independence number via theϑ-function , 1998, Math. Program..

[13]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[14]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[15]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[16]  Colin McDiarmid,et al.  Topics in Chromatic Graph Theory: Colouring random graphs , 2015 .

[17]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[18]  Noga Alon,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1997, SIAM J. Comput..

[19]  Ari Juels,et al.  Hiding Cliques for Cryptographic Security , 1998, SODA '98.

[20]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1990, BIT.

[21]  Shang-Hua Teng,et al.  Spectral partitioning works: planar graphs and finite element meshes , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[22]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[23]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .