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19
Finding a large hidden clique in a random graph
, 1998
"... ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., 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 polynomia ..."
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Cited by 82 (5 self)
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ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., 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�cn0.5 ˇ, for
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 46 (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...
Algorithmic barriers from phase transitions. preprint
"... For many random Constraint Satisfaction Problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomialtime algorithms stop finding solutions at much smaller densities. For exa ..."
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Cited by 26 (4 self)
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For many random Constraint Satisfaction Problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomialtime algorithms stop finding solutions at much smaller densities. For example, it is wellknown that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 − ɛ)χ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of kcolorings of a random graph, viewed as a subset of {1,..., k} n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of kcolorings looks like a giant ball for k ≥ 2χ, but like an errorcorrecting code for k ≤ (2 − ɛ)χ. We also prove that an analogous phase transition occurs both in random kSAT and in random hypergraph 2coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomialtime algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1step ReplicaSymmetryBreaking hypothesis of statistical physics for random CSPs.
The probable value of the LovaszSchrijver relaxations for maximum independent set
 SIAM Journal on Computing
, 2003
"... independent set ..."
Testing kwise and almost kwise independence
 In 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In this work, we consider the problems of testing whether a distribution over {0, 1} n is kwise (resp. (ɛ, k)wise) independent using samples drawn from that distribution. For the problem of distinguishing kwise independent distributions from those that are δfar from kwise independence in statis ..."
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Cited by 20 (8 self)
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In this work, we consider the problems of testing whether a distribution over {0, 1} n is kwise (resp. (ɛ, k)wise) independent using samples drawn from that distribution. For the problem of distinguishing kwise independent distributions from those that are δfar from kwise independence in statistical distance, we upper bound the number of required samples by Õ(nk /δ 2) and lower bound it by Ω(n k−1 2 /δ) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution’s distance from kwise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest. To distinguish (ɛ, k)wise independent distributions from those that are δfar from (ɛ, k)wise independence in statistical distance, we upper bound the number of required samples by O ` k log n δ2ɛ2 ´ and lower bound it by
Public Key Cryptography from Different Assumptions
, 2008
"... We construct a new public key encryption based on two assumptions: 1. One can obtain a pseudorandom generator with small locality by connecting the outputs to the inputs using any sufficiently good unbalanced expander. 2. It is hard to distinguish between a random graph that is such an expander and ..."
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Cited by 10 (2 self)
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We construct a new public key encryption based on two assumptions: 1. One can obtain a pseudorandom generator with small locality by connecting the outputs to the inputs using any sufficiently good unbalanced expander. 2. It is hard to distinguish between a random graph that is such an expander and a random graph where a (planted) random logarithmicsized subset S of the outputs is connected to fewer than S  inputs. The validity and strength of the assumptions raise interesting new algorithmic and pseudorandomness questions, and we explore their relation to the current stateofart. 1
Scalable architecture for adiabatic quantum computing of NPhard problems
 In Quantum computing & Quantum Bits in Mesoscopic System
, 2003
"... We present a comprehensive review of past research into adiabatic quantum computation and then propose a scalable architecture for an adiabatic quantum computer that can treat NPhard problems without requiring local coherent operations. Instead, computation can be performed entirely by adiabaticall ..."
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Cited by 8 (0 self)
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We present a comprehensive review of past research into adiabatic quantum computation and then propose a scalable architecture for an adiabatic quantum computer that can treat NPhard problems without requiring local coherent operations. Instead, computation can be performed entirely by adiabatically varying a magnetic field applied to all the qubits simultaneously. Local (incoherent) operations are needed only for: (1) switching on or off certain pairwise, nearestneighbor inductive couplings in order to set the problem to be solved and (2) measuring some subset of the qubits in order to obtain the answer to the problem.
Coloring Random Graphs  an Algorithmic Perspective
 In Proceedings of the 2nd Colloquium on Mathematics and Computer Science (MathInfo’2002
, 2002
"... Algorithmic Graph Coloring and Random Graphs have long become one of the most prominent branches of Combinatorics and Combinatorial Optimization. It is thus very natural to expect that their mixture will produce quite many very attractive, diverse and challenging problems. And indeed, the last thirt ..."
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Cited by 6 (0 self)
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Algorithmic Graph Coloring and Random Graphs have long become one of the most prominent branches of Combinatorics and Combinatorial Optimization. It is thus very natural to expect that their mixture will produce quite many very attractive, diverse and challenging problems. And indeed, the last thirty or so years witnessed rapid growth of the field of Algorithmic Random Graph Coloring, with many researchers working in this area and bringing there their experience from different directions of Combinatorics, Probability and Computer Science. One of the most distinctive features of this field is indeed the diversity of tools and approaches used to tackle its central problems. This survey is not intended to be a very detailed, monographlike coverage of Algorithmic Random Graph Coloring. Instead, our aim is to acquaint the reader with several of the main problems in the field and to show several of the approaches that proved most fruitful in attacking those problems. We do not and we simply cannot...
Small clique detection and approximate Nash equilibria
 In 13th International Workshop on Randomization and Computation (RANDOM
, 2009
"... Abstract. Recently, Hazan and Krauthgamer showed [12] that if, for a fixed small ε, an εbest εapproximate Nash equilibrium can be found in polynomial time in twoplayer games, then it is also possible to find a planted clique in Gn,1/2 of size C log n, where C is a large fixed constant independent ..."
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Cited by 4 (0 self)
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Abstract. Recently, Hazan and Krauthgamer showed [12] that if, for a fixed small ε, an εbest εapproximate Nash equilibrium can be found in polynomial time in twoplayer games, then it is also possible to find a planted clique in Gn,1/2 of size C log n, where C is a large fixed constant independent of ε. In this paper, we extend their result to show that if an εbest εapproximate equilibrium can be efficiently found for arbitrarily small ε> 0, then one can detect the presence of a planted clique of size (2+δ) log n in Gn,1/2 in polynomial time for arbitrarily small δ> 0. Our result is optimal in the sense that graphs in Gn,1/2 have cliques of size (2 − o(1)) log n with high probability. 1
A Sharp PageRank Algorithm with Applications to Edge Ranking and Graph Sparsification
, 2010
"... We give an improved algorithm for computing personalized PageRank vectors with tight error bounds which can be as small as Ω(n −p) for any fixed positive integer p. The improved PageRank algorithm is crucial for computing a quantitative ranking of edges in a given graph. We will use the edge rankin ..."
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Cited by 3 (3 self)
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We give an improved algorithm for computing personalized PageRank vectors with tight error bounds which can be as small as Ω(n −p) for any fixed positive integer p. The improved PageRank algorithm is crucial for computing a quantitative ranking of edges in a given graph. We will use the edge ranking to examine two interrelated problems – graph sparsification and graph partitioning. We can combine the graph sparsification and the partitioning algorithms using PageRank vectors to derive an improved partitioning algorithm.