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44
Random Walks on Truncated Cubes and Sampling 01 Knapsack Solutions
, 2003
"... We solve an open problem concerning the mixing time of symmetric random walk on the ndimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsack p ..."
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Cited by 50 (1 self)
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We solve an open problem concerning the mixing time of symmetric random walk on the ndimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsack problem. The results extend to the case of any fixed number of hyperplanes. The key ingredient in our analysis is a combinatorial construction we call a "balanced almost uniform permutation," which is of independent interest.
Every linear threshold function has a lowweight approximator
, 2006
"... Given any linear threshold function f on n Boolean variables, we construct a linear thresholdfunction g which disagrees with f on at most an ffl fraction of inputs and has integer weights each of magnitude at most pn * 2 ~O(1/ffl 2). We show that the construction is optimal in terms of its dependen ..."
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Cited by 36 (14 self)
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Given any linear threshold function f on n Boolean variables, we construct a linear thresholdfunction g which disagrees with f on at most an ffl fraction of inputs and has integer weights each of magnitude at most pn * 2 ~O(1/ffl 2). We show that the construction is optimal in terms of its dependence on n by proving a lower bound of \Omega (pn) on the weights required to approximatea particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately countingthe fraction of satisfying assignments to an instance of the zeroone knapsack problem to within an additive +ffl. The algorithm runs in time polynomial in n (but exponential in 1/ffl2).In our second application, we show that any linear threshold function f is specified to withinerror ffl by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive +1/(n * 2 ~O(1/ffl 2)). This is the first such accuracy bound which is inverse polynomial in n (previous work of Goldberg [12] gave a 1/quasipoly(n) bound), andgives the first polynomial bound (in terms of
Perfect simulation and backward coupling
 Comm. Statist. Stochastic Models
, 1998
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Mathematical foundations of the Markov chain Monte Carlo method
 in Probabilistic Methods for Algorithmic Discrete Mathematics
, 1998
"... 7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that a ..."
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Cited by 34 (1 self)
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7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction 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 = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12
Rapidly Mixing Markov Chains with Applications in Computer Science and Physics
, 2006
"... Monte Carlo algorithms often depend on Markov chains to sample from very large data sets. A key ingredient in the design of an efficient Markov chain is determining rigorous bounds on how quickly the chain “mixes,” or converges, to its stationary distribution. This survey provides an overview of sev ..."
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Cited by 33 (0 self)
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Monte Carlo algorithms often depend on Markov chains to sample from very large data sets. A key ingredient in the design of an efficient Markov chain is determining rigorous bounds on how quickly the chain “mixes,” or converges, to its stationary distribution. This survey provides an overview of several useful techniques.
Testing Closeness of Discrete Distributions
"... Given samples from two distributions over an nelement set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in n, specifically, O(n 2/3 ǫ −8/3 log n), independent samples from each distribution, runs in time linear in the sample size, ..."
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Cited by 26 (1 self)
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Given samples from two distributions over an nelement set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in n, specifically, O(n 2/3 ǫ −8/3 log n), independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than max{ǫ 4/3 n −1/3 /32, ǫn −1/2 /4}) or large (more than ǫ) in ℓ1 distance. This result can be compared to the lower bound of Ω(n 2/3 ǫ −2/3) for this problem given by Valiant [2008]. Our algorithm has applications to the problem of testing whether a given Markov process is rapidly mixing. We present sublinear algorithms for several variants of this problem as well.
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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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...
Random walks on combinatorial objects
 Surveys in Combinatorics 1999
, 1999
"... Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the unde ..."
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Cited by 22 (8 self)
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Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem. 1
Delayed path coupling and generating random permutations via distributed stochastic processes
, 1999
"... We analyze various stochastic processes for generating permutations almost uniformly at random in distributed and parallel systems. All our protocols are simple, elegant and are based on performing disjoint transpositions executed in parallel. The challenging problem of our concern is to prove that ..."
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Cited by 18 (3 self)
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We analyze various stochastic processes for generating permutations almost uniformly at random in distributed and parallel systems. All our protocols are simple, elegant and are based on performing disjoint transpositions executed in parallel. The challenging problem of our concern is to prove that the output configurations in our processes reach almost uniform probability distribution very rapidly, i.e. in a (low) polylogarithmic time. For the analysis of the aforementioned protocols we develop a novel technique, called delayed path coupling, for proving rapid mixing of Markov chains. Our approach is an extension of the path coupling method of Bubley and Dyer. We apply delayed path coupling to three stochastic processes for generating random permutations. For one
Coupling vs. Conductance for the JerrumSinclair Chain
, 1999
"... We address the following question: is the Causal Coupling method as strong as the Conductance method in showing rapid mixing of Markov Chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negat ..."
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Cited by 12 (0 self)
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We address the following question: is the Causal Coupling method as strong as the Conductance method in showing rapid mixing of Markov Chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the JerrumSinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. An extended abstract of this work appeared in the Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, 1999.