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57
The objective method: Probabilistic combinatorial optimization and local weak convergence
, 2003
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A proof of Parisi’s conjecture on the random assignment problem
 PROBAB. THEORY RELAT. FIELDS
, 2003
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Counting without sampling. New algorithms for enumeration problems using statistical physics
 IN PROCEEDINGS OF SODA
, 2006
"... We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in stati ..."
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Cited by 29 (8 self)
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We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ǫapproximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4regular nnode graph with large girth has approximately (1.494...) n independent sets, and in every rregular graph with n nodes and large girth the number of q ≥ r + 1proper colorings is approximately [q(1 − 1 r q) 2] n, for large n. In statistical physics terminology, we compute explicitly the limit of the logpartition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.
Rounding of continuous random variables and oscillatory asymptotics
 Ann. Probab
"... We study the characteristic function and moments of the integervalued random variable ⌊X + α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integerva ..."
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Cited by 18 (6 self)
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We study the characteristic function and moments of the integervalued random variable ⌊X + α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integervalued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries. 1. Introduction. Let
The number of matchings in random graphs
 Journal of Statistical Mechanics: Theory and Experiment
"... Abstract. We study matchings on sparse random graphs by means of the cavity method. ..."
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Cited by 16 (0 self)
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Abstract. We study matchings on sparse random graphs by means of the cavity method.
A rigorous proof of the cavity method for counting matchings
"... In this paper we rigorously prove the validity of the cavity method for the problem of counting the number of matchings in graphs with large girth. Cavity method is an important heuristic developed by statistical physicists that has lead to the development of faster distributed algorithms for probl ..."
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Cited by 13 (4 self)
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In this paper we rigorously prove the validity of the cavity method for the problem of counting the number of matchings in graphs with large girth. Cavity method is an important heuristic developed by statistical physicists that has lead to the development of faster distributed algorithms for problems in various combinatorial optimization problems. The validity of the approach has been supported mostly by numerical simulations. In this paper we prove the validity of cavity method for the problem of counting matchings using rigorous techniques. We hope that these rigorous approaches will finally help us establish the validity of the cavity method in general.
On the value of a random minimum weight Steiner tree
 Combinatorica
, 2004
"... Consider a complete graph on n vertices with edge weights chosen randomly and independently from, for example, an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of ..."
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Cited by 11 (0 self)
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Consider a complete graph on n vertices with edge weights chosen randomly and independently from, for example, an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1 + o(1))(k 1)(log n log k)=n when k = o(n) and n !1.
Random assignment and shortest path problems
, 2006
"... We explore a similarity between the n by n random assignment problem and the random shortest path problem on the complete graph on n + 1 vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We g ..."
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Cited by 10 (2 self)
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We explore a similarity between the n by n random assignment problem and the random shortest path problem on the complete graph on n + 1 vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his ζ(2) limit theorem for the assignment problem.