Alon, N., Krivelevich, M., and Sudakov, B. Finding a large hidden clique in a random graph. Random Structures Algorithms 13, 3-4 (1998), 457-466.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... This dual role played by randomness is well studied in statistical mechanics models, e.g. in the study of the spin glass model and its variants [6] and is closely related to other planted NP instances such as planted clique and planted bisection, that have received much attention in recent years [4, 15, 18]. For this distribution we have negative results. We show that for large enough constant density, almost all formulas are hard for RWalkSAT, and require exponential time until a satisfying assignment is found (theorem 4.2) In order to prove this, we rst investigate the full CNF, comprising of ....

....[6] we propose to look at the following pair of planted SAT distributions over satis able 3 CNFs. This distribution is highly interesting in its own right. It has been studied empirically in [9] and is the analog of the planted clique and planted bisection distributions, studied e.g. in [4, 15, 18]. De nition 4.1 (Planted SAT) Let S be the distribution obtained by selecting at random 2 f0; 1g , and selecting at random n clauses out of all clauses of size 3 that are satis ed by . Denote a random formula from this distribution by C S . Let P be the distribution ....

N. Alon, M. Krivelevich, B. Sudakov. Finding a large hidden clique in a random graph, In Random Structures and Algorithms 13 (1998), 457-466.

....could be cryptography. The objective here is to find a large clique hidden (placed randomly) in a random graph. If it is hard to find a clique, and it remains hard to find a hidden clique, then it could be used in cryptography instead of prime factorization. Alon, Krivelevich and Sudakov (1998)[11] study the hardness of finding a large hidden clique in a random graph. They present an e#cient (polynomial) algorithm to find almost surely, for all k cn 0.5 , for any fixed c 0, a hidden fixed clique of size k in a random graph G(n, 1 2) 23 Therefore Juels and Peinado (1998) 145] are ....

N. Alon, M. Krivelevich and B. Sudakov, Finding a large Hidden Clique in a random graph, Random Struct. Alg. 13 (

....graph a clique of size k. For the hidden clique problem in the random model, Kucera [Kuc95] observed that taking the vertices with highest degrees almost surely succeeds in finding the hidden clique, when k c p n log n for a sufficiently large constant c 0. Alon, Krivelevich and Sudakov [AKS98] showed an algorithm based on eigenvalue techniques that almost surely finds the hidden clique, when k Omega Gamma p n) Jerrum showed that the Metropolis process does not find the clique, almost surely, when k = o( p n) We devise another heuristic for the hidden clique problem. Our ....

....clique, almost surely, when k Omega Gamma p n) but it extends to a semi random model of the problem, in which an adversary is allowed to remove (from the random graph with 10 the planted clique) any edge that is not inside the planted clique. In contrast, the previous algorithms of [Kuc95, AKS98] have similar success in the random model, but fail in the semirandom model, unless k = Omega Gamma n) An additional useful property of our heuristic is that it almost surely certifies the optimality its solution. Namely, the heuristic produces its solution together with an upper bound on the ....

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N. Alon, M. Krivelevich, and B. Sudakov. Finding a large hidden clique in a random graph. Random Structures Algorithms, 13(3-4):457--466, 1998.

....using a spectral algorithm. The problem of finding a large clique in a random graph was suggested by Karp in [16] Kucera [19] observes that when the size of the clique is ( p n log n) and p = 1=2, the clique members are simply the vertices of highest degree. Alon, Krivelevich, and Sudakov [3] showed that a planted clique of size p n) can be found through spectral techniques when p = 1=2. 1.1 Graph Partition Model The graph models above each generate a graph by including each edge independently with an associated probability. Further, in each model we can partition the nodes of the ....

....c. Corollary 3 Let ( p) be an instance of the planted clique problem, where the clique size is s. There is a constant c such that for sufficiently large n if 1 p p c n s 2 log(n= s then we can recover with probability 1 . This result subsumes the spectral result of [3] where they show that cliques of size p n) can be found when p = 1=2. Note that this theorem allows for variable p and s . The restriction to a single clique is also not necessary. The general theorem addresses graphs with several hidden cliques and hidden independent sets, each of varying ....

Noga Alon, Michael Krivelevich, and Benny Sudakov, Finding a large hidden clique in a random graph, Random Structures and Algorithms 13 (1998), 457--466.

....of the match: 2n Hamming distance to the linear bisection) To form our four quadrants this polar space is divided into up, down, left, and right quadrants. By mirror image symmetry the up and down quadrants should be equal fractions of the search space, as should the lef t and right. Figure 2 illustrates that these symmetries held for our samples. We will also use these metrics later when mapping the search space. The GWW algorithm at the given parameter settings should be stable: measurements should not change if the parameters are increased. We used stability in two tests that the ....

....the delicate nature of the balance and never contradicted any of our conjectures about the search space. Graph Alg. Mean Batch bMean Found planted GR 258.6 25 198.2 11 mGR 215.0 8 190.4 34 400 BA 206.7 6 189.5 53 vertices oe 193.3 3 189.0 98 GWW 190.9 5 189.2 82 planted GR 14047 2 13954 0 mGR 11631 6 9327 100 20000 BA 11536 6 9323 100 vertices oe 10165 20 9320 100 GWW 9376 3 9331 100 geometric GR 1030.4 177 482.3 14 mGR 850.2 23 484.6 22 500 BA 733.8 11 484.5 42 vertices oe 487.1 1 468.9 36 GWW 468.9 1 487.1 66 Table 1: Performance of algorithms on ....

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Alon, N., M. Krivelevich, B. Sudakev. Finding al large hidden clique in a random graph, SIAM Symposium on Discrete Algorithms, Jan. 1998, pp. 594-598.

....independent set; for reasonably large K and p, S is the unique largest independent set. The larger K is, the easier it is to find the independent set. For a constant p, the lowest value of K that can be provably handled by known heuristics (spectral techniques, in this case) is Omega Gamma p n) [4]. In the random k colorable graph model, a graph is partitioned into k color classes, and edges are placed at random with probability p between color classes. Alon and Kahale [2] also using spectral methods, show that with high probability a k coloring can be recovered when p c=n and c 0 is a ....

N. Alon, M. Krivelevich and B. Sudakov. "Finding a large hidden clique in a random graph". Random Structures and Algorithms 13 (1998), 457--466.

....graph models. Kucera [22] considers a variety of graph partitioning problems, including bisection and coloring, on random graphs. He suggests algorithms for solving these with high probability and studies their expected complexity assuming some specific distributions on the input. Alon et. al [1] give an algorithm for finding a single large hidden clique in a random graph. Condon and Karp [6] consider the graph l partition problem: partition the nodes of an undirected graph into l subsets of predefined sizes so that the total number of inter subset edges is minimal. They present an ....

N. Alon, M. Krivelevich, and B. Sudakov. Finding a large hidden clique in a random graph. In Proc. Ninth Annual ACM-SIAM Sympoium on Discrete Algorithms (SODA 98), pages 594--598, San Francisco, California, 1998. 22

....not find the clique when k = o( p n) Kucera observed that when k c p n log n for an appropriate constant c, the vertices of the planted clique would almost surely be the ones with the largest degrees in G, and hence it is easy to recognize them efficiently. Alon, Krivelevich and Sudakov [AKS98] showed an algorithm based on spectral properties of the graph, for finding the planted clique, almost surely, when k Omega Gamma p n) A major motivation for studying various probabilistic input models in general is to evaluate algorithms performance in real life applications. It would be ....

....are equal, almost surely, in which case the algorithm proves the optimality of its output bisection. We present an algorithm based on the Lov asz theta function for finding a clique of size k Omega Gamma p n) planted in a random graph G n;1=2 . Our algorithm improves over the algorithm of [AKS98] in two respects: 1. Extends to the sandwich model. An adversary may remove edges from the [AKS98] graph except for the edges forming the clique of size k. In the sandwich model terminology , let G max = G n;1=2;k be the [AKS98] graph, and let G min be the empty graph except the same clique of ....

[Article contains additional citation context not shown here]

Noga Alon, Michael Krivelevich, and Benny Sudakov. Finding a large hidden clique in a random graph. Random Structures and Algorithms, 13(3-4):457-- 466, 1998.

....not find the clique when k = o( p n) Kucera observed that when k c p n log n for an appropriate constant c, the vertices of the planted clique would almost surely be the ones with the largest degrees in G, and hence it is easy to recognize them efficiently. Alon, Krivelevich and Sudakov [2] showed an algorithm based on spectral properties of the graph, for finding the planted clique, almost surely, when k Omega Gamma p n) A major motivation for studying various probabilistic input models in general is to evaluate algorithms performance in real life applications. It would be ....

....are equal, almost surely, in which case the algorithm proves the optimality of its output bisection. We present an algorithm based on the Lov asz theta function for finding a clique of size k Omega Gamma p n) planted in a random graph G n;1=2 . Our algorithm improves over the algorithm of [2] in two respects: 1. Extends to the sandwich model. An adversary may remove edges from the [2] graph except for the edges forming the clique of size k. In the sandwich model terminology , let G max = G n;1=2;k be the [2] graph, and let G min be the empty graph except the same clique of size k. ....

[Article contains additional citation context not shown here]

Noga Alon, Michael Krivelevich, and Benny Sudakov. Finding a large hidden clique in a random graph. Random Structures and Algorithms, 13(3-4):457--466, 1998.

....is impossible unless a tractable algorithm can solve an NPC problem. Thus, we can not expect to construct an approximation algorithm that guarantees good performance. The best known approximation algorithm, designed by Boppana and Halld rsson, has a performance guarantee of O # n (log n) 2 # [3]. Due to its NPC status, nding correct solutions to the clique problem is assumed to be hard. Tarjan and Trojanowski develop a recursive O # 2 n 3 # algorithm that solves the clique problem correctly [4] Our objective is to provide an approximation algorithm that works well on the average, ....

Alon, Krivelevich, and Sudakov, Finding a Large Hidden Clique in a Random Graph, Proceedings of the Ninth Annual ACM-SIAM SODA, 1998, pp. 594598.

....at random is hard, it is not obvious that the problem remains hard after a clique is embedded in G. This is because the distribution of graphs induced by embedding a clique is no longer uniform. For example, when k n 1=2 ffl (n = jV j; ffl 0) it is easy to find the hidden clique. See also [1] in this volume. Our aim is to show that k can be chosen such that this is not the case. Our main result shows for suitable k that finding the hidden clique or any other clique in G with k nodes is at least as hard as finding a clique with k nodes under the uniform distribution. We believe ....

N. Alon, M. Krivelevich, and B. Sudakov. Finding a large hidden clique in a random graph. In Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998.

....the density of the random graphs under consideration. There is also another result [55] about the smallest maximal clique and it shows that its size is almost surely M(n; ffi) 2 in the limit of large n. Another interesting result for these graphs is known as the Jerrum conjecture [188] see also [10] for recent, related results) It states that in large random graphs of density ffi = 1=2 there is no polynomial time algorithm that, with probability greater than 1=2, can find a clique larger than the smallest maximal clique. 5 Exact Algorithms 5.1 Enumerative Algorithms The first algorithm ....

N. Alon, M. Krivelevich and B. Sudakov, Finding a large hidden clique in a random graph, in: Proc. SODA '98---Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco CA, January 25-- 27, 1998.

....with understanding of the information provided by these parameters can constitute a very powerful tool, capable of solving algorithmic problems where all other methods failed. This is especially true for randomly generated graphs, several successful examples of spectral techniques are [6] 2] [3]. A survey [1] discusses several applications of spectral techniques to graph algorithms. In order to show that bad graphs have an exponentially small probability in G(n; p) we will prove a new large deviation result for eigenvalues of random symmetric matrices. This result, bounding the tails ....

....pick randomly a subset Q of size k and force it to be independent by erasing all edges inside Q. We denote the last model of random graphs by G(n; 1=2; k) The problem is to distinguish in polynomial time between the above two models. For the case k = n 1=2 ) Alon, Krivelevich and Sudakov [3] showed how to recover the independent set of size k in G(n; 1=2; k) using spectral techniques, thus clearly providing a tool for distinguishing between G(n; 1=2) and G(n; 1=2; k) See also [10] for a related result. However, Saks question is still open for every k = o(n 1=2 ) Returning to ....

N. Alon, M. Krivelevich and B. Sudakov, Finding a large hidden clique in a random graph, Proc. 9 th ACM-SIAM SODA, ACM Press (1998), 594-598.

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Alon, N., Krivelevich, M., and Sudakov, B. Finding a large hidden clique in a random graph. Random Structures Algorithms 13, 3-4 (1998), 457-466.

No context found.

N. Alon, M.Krivelevich, B. Sudakov. Finding a large hidden clique in a random graph, In Random Structures and Algorithms 13 (1998), 457-466.

No context found.

N. Alon, M. Krivelevich. and B. Sudakev. Finding a large hidden clique in a random graph. In Proceedings of the 9th Annual SIAM Symposium on Discrete Algorithms, 1998, pp. 594-598.

No context found.

N. Alon, M. Krivelevich, and B. Sudakov. Finding a large hidden clique in a random graph. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 594-598, 1998.

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Noga Alon, Michael Krivelevich, and Benny Sudakov, Finding a large hidden clique in a random graph, 9th Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA,

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