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42
The Markov Chain Monte Carlo method: an approach to approximate counting and integration
, 1996
"... In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stocha ..."
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Cited by 234 (12 self)
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In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of nonasymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.
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 80 (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
A Column Generation Approach For Graph Coloring
 INFORMS Journal on Computing
, 1995
"... We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approac ..."
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Cited by 71 (2 self)
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We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...
What do we know about the Metropolis algorithm
 J. Comput. System. Sci
, 1998
"... The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an il ..."
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Cited by 61 (7 self)
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The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains. 1. Introduction. Let % be a finite set and m(~)> 0 a probability distribution on %. The Metropolis algorithm is a procedure for drawing samples from X. It was introduced by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller [1953]. The algorithm requires the user to specify a connected, aperiodic Markov chain 1<(z, y) on %. This chain need not be symmetric but if K(z, y)>0, one needs 1<(Y, z)>0. The chain K is modified
On Counting Independent Sets in Sparse Graphs
, 1998
"... We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if ..."
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Cited by 58 (11 self)
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We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if \Delta 25, unless RP = NP. 1 Introduction Counting independent sets in graphs is one of several combinatorial counting problems which have received recent attention. The problem is known to be #Pcomplete, even for low degree graphs [3]. On the other hand, it has been shown that, for graphs of maximum degree \Delta = 4, randomized approximate counting is possible [7, 3]. This success has been achieved using the Monte Carlo Markov chain method to construct a fully polynomial randomized approximation scheme (fpras). This has led to a natural question as to how far this success might extend. Here we consider in more detail this question of counting independent sets in graphs with constant m...
Simulated annealing for graph bisection
 in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science
, 1993
"... We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in Gnpr when p r = O(n*’) for A 5 2. (The random graph model Gnpr specifies a “planted ” bisection of density r, ..."
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Cited by 35 (1 self)
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We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in Gnpr when p r = O(n*’) for A 5 2. (The random graph model Gnpr specifies a “planted ” bisection of density r, separating two n/2vertex subsets of slightly higher density p.) We show that simulated “annealing ” at an appropriate fixed temperature (i.e., the Metropolis algorithm) finds the unique smallest bisection in O(n2+‘) steps with very high probability, provided A> 1116. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n’+‘).) We leave open the question of whether annealing is effective for A in the range 312 < A 5 1116, whose lower limit represents the threshold at which the planted bisection becomes lost amongst other random small bisections. It also remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem. 1
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 23 (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
Spectral Techniques in Graph Algorithms
 Lecture Notes in Computer Science 1380
, 1998
"... The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs. ..."
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Cited by 23 (2 self)
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The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs. 1