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A PolynomialTime Approximation Algorithm for the Permanent of a Matrix with NonNegative Entries
 Journal of the ACM
, 2004
"... Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily ..."
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Cited by 438 (26 self)
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Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical
NonDeterministic Exponential Time has TwoProver Interactive Protocols
"... We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language ..."
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Cited by 437 (38 self)
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We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language L. It was previously suspected (and proved in a relativized sense) that coNPcomplete languages do not admit such proof systems. In sharp contrast, we show that the class of languages having twoprover interactive proof systems is nondeterministic exponential time. After the recent results that all languages in PSPACE have single prover interactive proofs (Lund, Fortnow, Karloff, Nisan, and Shamir), this represents a further step demonstrating the unexpectedly immense power of randomization and interaction in efficient provability. Indeed, it follows that multiple provers with coins are strictly stronger than without, since NEXP # NP. In particular, for the first time, provably polynomial time intractable languages turn out to admit “efficient proof systems’’ since NEXP # P. We show that to prove membership in languages in EXP, the honest provers need the power of EXP only. A consequence, linking more standard concepts of structural complexity, states that if EX P has polynomial size circuits then EXP = Cg = MA. The first part of the proof of the main result extends recent techniques of polynomial extrapolation of truth values used in the single prover case. The second part is a verification scheme for multilinearity of an nvariable function held by an oracle and can be viewed as an independent result on program verification. Its proof rests on combinatorial techniques including the estimation of the expansion rate of a graph.
A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents
"... ..."
Linear Assignment Problems and Extensions
"... This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems ..."
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Cited by 66 (0 self)
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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial threedimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published
A deterministic approximation algorithm for computing a permanent of a 0, 1 matrix
, 2008
"... We construct a deterministic approximation algorithm for computing a permanent of a 0, 1 n by n matrix to within a multiplicative factor (1 + ǫ) n, for arbitrary ǫ> 0. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general c ..."
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Cited by 51 (9 self)
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We construct a deterministic approximation algorithm for computing a permanent of a 0, 1 n by n matrix to within a multiplicative factor (1 + ǫ) n, for arbitrary ǫ> 0. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is exp(O(n 2 3 log 3 n)). For the class of graphs which are constant degree expanders the first result is an improvement over the best known approximation factor e n obtained in [LSW00]. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph [BGK +] and JerrumVazirani expander decomposition method of [JV96].
Arithmetization: A New Method In Structural Complexity Theory
, 1991
"... . We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes via multivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes there ..."
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Cited by 50 (9 self)
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. We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes via multivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes thereby available for the study of complexity classes. The theory to be described provides a unified framework from which powerful recent results follow naturally. The central result is a characterization of ]P in terms of arithmetic straight line programs. The consequences include a simplified proof of Toda's Theorem that PH ` P ]P ; and an infinite class of natural and potentially inequivalent functions, checkable in the sense of Blum et al. Similar characterizations of PSPACE are also given. The arithmetization technique was independently discovered by Adi Shamir. While this simultaneous discovery was driven by applications to interactive proofs, the present paper demonstrates the applicabil...
Relationships Among PL, L, and the Determinant
, 1996
"... Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterizati ..."
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Cited by 37 (9 self)
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Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspacebounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.
Structured prediction via the extragradient method
"... We present a simple and scalable algorithm for largemargin estimation of structured models, including an important class of Markov networks and combinatorial models. The estimation problem can be formulated as a quadratic program (QP) that exploits the problem structure to achieve polynomial number ..."
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Cited by 31 (2 self)
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We present a simple and scalable algorithm for largemargin estimation of structured models, including an important class of Markov networks and combinatorial models. The estimation problem can be formulated as a quadratic program (QP) that exploits the problem structure to achieve polynomial number of variables and constraints. However, offtheshelf QP solvers scale poorly with problem and training sample size. We recast the formulation as a convexconcave saddle point problem that allows us to use simple projection methods. We show the projection step can be solved using combinatorial algorithms for mincost convex flow. We provide linear convergence guarantees for our method and present experiments on two very different structured prediction tasks: 3D image segmentation and word alignment, illustrating the favorable scaling properties of our algorithm.
Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... We present an improved “cooling schedule ” for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the ..."
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Cited by 31 (2 self)
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We present an improved “cooling schedule ” for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature (difficult region). We present applications of our technique to colorings and the permanent (perfect matchings of bipartite graphs). Moreover, for the permanent, we improve the analysis of the Markov chain underlying the simulated annealing algorithm. This improved analysis, combined with the faster cooling schedule, results in an O(n 7 log 4 n) time algorithm for approximating the permanent of a 0/1 matrix. 1
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.