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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 252 (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.
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 33 (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
Averagecase Analysis of Algorithms for Matchings and Related Problems
 Journal of the ACM
, 1994
"... We analyze the behavior of augmenting paths in random graphs. Our results show that in almost every graph, any nonmaximum 01 flow admits a short augmenting path. This enables us to prove that augmenting path algorithms, which are fast in the worst case, also perform exceedingly well on the average ..."
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Cited by 25 (0 self)
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We analyze the behavior of augmenting paths in random graphs. Our results show that in almost every graph, any nonmaximum 01 flow admits a short augmenting path. This enables us to prove that augmenting path algorithms, which are fast in the worst case, also perform exceedingly well on the average. In particular, we show that the O(&radic;(V) E) algorithms for bipartite and general matchings run in almost linear time with high probability. It is also shown that the expected running time of the matching algorithms is O(E) on input graphs chosen uniformly at random from the set of all graphs. We establish that the permanent of almost every bipartite graph can be approximated in polynomial time. We extend our results to the analysis of the running time of Dinic's algorithm for finding factors of graphs.
Approximating the Number of MonomerDimer Coverings of a Lattice
 Journal of Statistical Physics
, 1996
"... The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temper ..."
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Cited by 18 (2 self)
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The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial time approximation algorithms for computing the number of coverings with any specified number of monomers in ddimensional rectangular lattices with periodic boundaries, for any fixed dimension d , and in twodimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in contrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do n...
Probabilistic Analysis of Network Flow Algorithms
 Mathematics of Operations Research
, 1995
"... This paper is concerned with the design and probabilistic analysis of algorithms for the maximumflow problem and capacitated transportation problems. These algorithms run in linear time and, under certain assumptions about the probability distribution of edge capacities, obtain an optimal solution w ..."
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Cited by 6 (1 self)
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This paper is concerned with the design and probabilistic analysis of algorithms for the maximumflow problem and capacitated transportation problems. These algorithms run in linear time and, under certain assumptions about the probability distribution of edge capacities, obtain an optimal solution with high probability. The design of our algorithms is based on the following general method, which we call the mimicking method, for solving problems in which some of the input data is deterministic and some is random with a known distribution: 1. Replace each random variable in the problem by its expectation; this gives a deterministic problem instance that has a special form, making it particularly easy to solve; 2. Solve the resulting deterministic problem instance; 3. Taking into account the actual values of the random variables, mimic the solution of the deterministic instance to obtain a nearoptimal solution to the original problem; 4. Finetune this suboptimal solution to obtain an o...
Stochastic Graphs Have Short Memory: Fully Dynamic Connectivity in PolyLog Expected Time
, 1995
"... This paper presents an average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions). To this end we introduce the model of stochastic graph processes (i.e. dynamically changing random graphs with random equiprobable edge insertions and deletions). ..."
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Cited by 4 (2 self)
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This paper presents an average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions). To this end we introduce the model of stochastic graph processes (i.e. dynamically changing random graphs with random equiprobable edge insertions and deletions). As the process continues indefinitely, all potential edge locations (in V \Theta V ) may be repeatedly inspected (and learned) by the algorithm. This learning of the structure seems to imply that traditional random graph analysis methods cannot be employed (since an observed edge is not a random event anymore). However, we show that a small (logarithmic) number of dynamic random updates are enough to allow our algorithm to reexamine edges as if they were random with respect to certain events (i.e. the graph "forgets" its structure). This short memory property of the stochastic graph process enables us to present an algorithm for graph connectivity which admits an amortized expected cost of...
AverageCase Analysis of GraphSearching Algorithms
, 1990
"... We estimate the expected value of various search quantities for a variety of graphsearching methods, for example depthfirst search and breadthfirst search. Our analysis applies to both directed and undirected random graphs, and it covers the range of interesting graph densities, including densit ..."
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Cited by 2 (0 self)
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We estimate the expected value of various search quantities for a variety of graphsearching methods, for example depthfirst search and breadthfirst search. Our analysis applies to both directed and undirected random graphs, and it covers the range of interesting graph densities, including densities at which a random graph is disconnected with a giant connected component. We estimate the number of edges examined during the search, since this number is proportional to the running time of the algorithm. We find that for hardly connected graphs, all of the edges might be examined, but for denser graphs many fewer edges are generally required. We prove that any searching algorithm examines \Theta(n log n) edges, if present, on all random graphs with n nodes but not necessarily on the complete graphs. One property of some searching algorithms is the maximum depth of the search. In depthfirst search, this depth can be used to estima...
Chapter 1 Sampling and Counting
"... The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = E(Z) of a random variable (r.v.) Z over a probability space Ω,µ. It is assumed that some efficient procedure for sampling from Ω,µ ..."
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The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = E(Z) of a random variable (r.v.) Z over a probability space Ω,µ. It is assumed that some efficient procedure for sampling from Ω,µ is available. By taking the mean of some
Bipartite Expander Matching is in NC
"... A workefficient deterministic NC algorithm is presented for finding a maximum matching in a bipartite expander graph with any expansion factor fi ? 1. This improves upon a recently presented deterministic NC maximum matching algorithm which is restricted to those bipartite expanders with large expa ..."
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A workefficient deterministic NC algorithm is presented for finding a maximum matching in a bipartite expander graph with any expansion factor fi ? 1. This improves upon a recently presented deterministic NC maximum matching algorithm which is restricted to those bipartite expanders with large expansion factors (fi \Delta ffl ; ffl ? 0), and is not workefficient [1]. Keywords: Bipartite Matching, Expander Graphs, NC, Network Flow. 1. Introduction Finding maximum cardinality matchings in bipartite expander graphs has many applications such as routing networks, sorting networks, permutation networks, and path selection. Note that by Hall's Theorem there is a perfect matching in a bipartite expander graph with expansion factor fi ? 1. Thus we are really finding one of the (potentially many) perfect matchings. Bipartite expander graphs are an important part of the design of routing networks such as concentrators and superconcentrators [2]. They are also used in selfrouting permuta...