Results 1  10
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28
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 (13 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.
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By d ..."
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Cited by 174 (28 self)
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We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.
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.
Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
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Cited by 24 (1 self)
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The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
Simulated Annealing for Convex Optimization
 Mathematics of Operations Research
, 2004
"... informs ® ..."
HillClimbing Finds Random Planted Bisections
 Proc. 12th Symposium on Discrete Algorithms (SODA 01), ACM press, 2001
, 2001
"... We analyze the behavior of hillclimbing algorithms for the minimum bisection problem on instances drawn from the "planted bisection" random graph model, Gn;p;q , previously studied in [3, 4, 10, 12, 15, 9, 7]. This is one of the few problem distributions for which various popular heuristic methods, ..."
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Cited by 11 (1 self)
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We analyze the behavior of hillclimbing algorithms for the minimum bisection problem on instances drawn from the "planted bisection" random graph model, Gn;p;q , previously studied in [3, 4, 10, 12, 15, 9, 7]. This is one of the few problem distributions for which various popular heuristic methods, such as simulated annealing, have been proven to succeed. However, it has been open whether these sophisticated methods were necessary, or whether simpler heuristics would also work. Juels [15] made the first progress towards an answer by showing that simple hillclimbing does suffice for very wide separations between p and q.
Finding a randomly planted assignment in a random 3CNF
 In preparation
, 2002
"... Abstract We discuss the problem of finding a satisfying assignment for a randomly chosen satisfiable 3 CNF with n variables and \Delta n clauses, where \Delta is much greater than the satisfiability threshold. We show that for \Delta = \Omega (log n), a trivial algorithm solves this problem. Our p ..."
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Cited by 8 (1 self)
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Abstract We discuss the problem of finding a satisfying assignment for a randomly chosen satisfiable 3 CNF with n variables and \Delta n clauses, where \Delta is much greater than the satisfiability threshold. We show that for \Delta = \Omega (log n), a trivial algorithm solves this problem. Our proof follows by showing that for large clause density, the distribution over random satisfiable 3CNFs is statistically close to the plantedSAT distribution. This latter distribution is much easier to analyze, and it is for this distribution that we present our algorithm. Keywords: Satisfiability, 3SAT, Random 3CNF, planted SAT.
Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model
"... In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model ..."
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Cited by 8 (0 self)
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In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degreecorrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing effect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degreecorrected randomwalk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model.
Towards an Analysis of Local Optimization Algorithms
 In Proc. 28th STOC
, 1996
"... We introduce a variant of Aldous and Vazirani's "Go with the winners" algorithm that can be used for search graphs that are not trees. We analyze the algorithm in terms of the properties of a treedecomposition of the search graph. We show a large class of distributions for search graphs so that "Go ..."
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Cited by 6 (1 self)
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We introduce a variant of Aldous and Vazirani's "Go with the winners" algorithm that can be used for search graphs that are not trees. We analyze the algorithm in terms of the properties of a treedecomposition of the search graph. We show a large class of distributions for search graphs so that "Go with the winners" works well with high probability for almost all graphs from the distribution. We also give a sufficient combinatorial property that ensures good performance.