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72
Clustering Gene Expression Patterns
, 1999
"... Recent advances in biotechnology allow researchers to measure expression levels for thousands of genes simultaneously, across different conditions and over time. Analysis of data produced by such experiments offers potential insight into gene function and regulatory mechanisms. A key step in the ana ..."
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Cited by 332 (11 self)
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Recent advances in biotechnology allow researchers to measure expression levels for thousands of genes simultaneously, across different conditions and over time. Analysis of data produced by such experiments offers potential insight into gene function and regulatory mechanisms. A key step in the analysis of gene expression data is the detection of groups of genes that manifest similar expression patterns. The corresponding algorithmic problem is to cluster multicondition gene expression patterns. In this paper we describe a novel clustering algorithm that was developed for analysis of gene expression data. We define an appropriate stochastic error model on the input, and prove that under the conditions of the model, the algorithm recovers the cluster structure with high probability. The running time of the algorithm on an ngene dataset is O(n 2 (log(n)) c ). We also present a practical heuristic based on the same algorithmic ideas. The heuristic was implemented and its p...
The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 140 (20 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
Spectral Partitioning of Random Graphs
, 2001
"... Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all ..."
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Cited by 87 (3 self)
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Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident.
Finding and Certifying a Large Hidden Clique in a SemiRandom Graph
, 1999
"... Alon, Krivelevich and Sudakov (Random Structures and Algorithms, 1998) designed an algorithm based on spectral techniques that almost surely finds a clique of size \Omega\Gamma p n) hidden in an otherwise random graph. We show that a different algorithm, based on the Lov'asz theta function, al ..."
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Cited by 47 (11 self)
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Alon, Krivelevich and Sudakov (Random Structures and Algorithms, 1998) designed an algorithm based on spectral techniques that almost surely finds a clique of size \Omega\Gamma p n) hidden in an otherwise random graph. We show that a different algorithm, based on the Lov'asz theta function, almost surely both finds the hidden clique and certifies its optimality. Our algorithm has an additional advantage of being more robust: it also works in a semirandom hidden clique model, in which an adversary can remove edges from the random portion of the graph. 1 Introduction A clique in a graph G is a subset of the vertices every two of which are connected by an edge. The maximum clique problem, that is, finding a clique of maximum size in a graph, is fundamental in the area of combinatorial optimization, and is closely related to the independent set problem (clique on the edge complement graph G), the vertex cover problem (the vertex complement of the independent set) and chromatic...
Heuristics for Semirandom Graph Problems
 Journal of Computing and System Sciences
, 2001
"... We consider semirandom graph models for finding large independent sets, colorings and bisections in graphs. These models generate problem instances by blending random and adversarial decisions. To generate semirandom independent set problems, an independent set S of ffn vertices is randomly chos ..."
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Cited by 36 (4 self)
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We consider semirandom graph models for finding large independent sets, colorings and bisections in graphs. These models generate problem instances by blending random and adversarial decisions. To generate semirandom independent set problems, an independent set S of ffn vertices is randomly chosen. Each edge connecting S with S is chosen with probability p, and an adversary is then allowed to add new edges arbitrarily, provided that S remains an independent set. The smaller p is, the greater the control the adversary has over the semirandom graph. We give a heuristic that with high probability recovers an independent set of size ffn whenever p ? (1 + ffl) ln n=ffn, for any constant ffl ? 0. We show that when p ! (1 \Gamma ffl) ln n=ffn, an independent set of size jSj cannot be recovered, unless NP ` BPP . We use our result for maximum independent sets to obtain greatly improved heuristics for the model of kcolorable semirandom graphs introduced by Blum and Spencer. For ...
Spectral techniques applied to sparse random graphs. Random Structures and Algorithms
 Random Structures and Algorithms
, 2003
"... We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ1 ≥... ≥ λn be the eigenvalues of an nvertex graph, and let λ = max[λ2, λn]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ1 ≥ d, and we show that λ = O ( √ ..."
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Cited by 34 (3 self)
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We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ1 ≥... ≥ λn be the eigenvalues of an nvertex graph, and let λ = max[λ2, λn]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ1 ≥ d, and we show that λ = O ( √ d). For d = c it is no longer true that λ = O ( √ d), but we show that by removing a small number of vertices of highest degree in G, one gets a graph G ′ for which λ = O ( √ d). Our proofs are based on the techniques of Kahn and Szemeredi from STOC 1989, who proved similar results for regular graphs. Our results are useful for extending the analysis of certain heuristics to sparser instances of NPhard problems. We illustrate this by removing some unnecessary logarithmic factors in the density of kSAT formulas that are refuted by the algorithm of Goerdt and Krivelevich from STACS 2001. 1
The probable value of the LovaszSchrijver relaxations for maximum independent set
 SIAM Journal on Computing
, 2003
"... independent set ..."
Approximating the independence number and the chromatic number in expected polynomial time
, 2001
"... ..."
Hiding Cliques for Cryptographic Security
 Des. Codes Cryptogr
, 1998
"... We demonstrate how a well studied combinatorial optimization problem may be introduced as a new cryptographic function. The problem in question is that of finding a "large" clique in a random graph. While the largest clique in a random graph is very likely to be of size about 2 log 2 n, it is widely ..."
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Cited by 26 (0 self)
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We demonstrate how a well studied combinatorial optimization problem may be introduced as a new cryptographic function. The problem in question is that of finding a "large" clique in a random graph. While the largest clique in a random graph is very likely to be of size about 2 log 2 n, it is widely conjectured that no polynomialtime algorithm exists which finds a clique of size (1 + ffl) log 2 n with significant probability for any constant ffl ? 0. We present a very simple method of exploiting this conjecture by "hiding" large cliques in random graphs. In particular, we show that if the conjecture is true, then when a large clique  of size, say, (1+2ffl) log 2 n  is randomly inserted ("hidden") in a random graph, finding a clique of size (1 + ffl) log 2 n remains hard. Our result suggests several cryptographic applications, such as a simple oneway function. 1 Introduction Many hard graph problems involve finding a subgraph of an input graph G = (V; E) with a certain propert...
Threshold Phenomena in Random Graph Colouring and Satisfiability
, 1999
"... We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k ..."
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Cited by 24 (4 self)
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We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k (n, d) = Pr[G(n, d/n) is kcolourable]. Erdos asked the following fundamental question: for k 3, is there a constant c k such that for any # > 0, #) = 1 , and lim f k (n, c k + #) = 0 ? (1) We prove that for all k 3, there exists a function t k (n) such that (1) holds upon replacing c k by t k (n), thus establishing that indeed kcolourability has a sharp threshold. Let d k = sup{d lim n## f k (n, d) = 1}. Note that if c k exists then, by definition, c k = d k . For the basic and most studied case k = 3 we prove 3.84 < d 3 < 5.05 . These are the best