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32
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 314 (23 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
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 231 (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.
The Complexity of Counting in Sparse, Regular, and Planar Graphs
 SIAM Journal on Computing
, 1997
"... We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to ..."
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Cited by 67 (0 self)
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We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. To achieve these results, a new interpolationbased reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NPhard even when restricted to graphs of maximal degree 3. Previously, restrictedcase complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems. 1 Introduction Ever since the introduction of NPcompleteness in the early 1970's, the primary focus of complexity theory has been on decision ...
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
"... ..."
Statistical Mechanics, ThreeDimensionality and NPcompleteness I. Universality of Intractability for the Partition Function of the Ising Model Across NonPlanar Lattices (Extended Abstract)
"... This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working ..."
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Cited by 28 (1 self)
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This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the CrossedBonds conjecture, and the impossibility to compute effectively the partition functions for any threedimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomialtime, paired with the results of this paper: for every NonPlanar crystal lattice computing the parition functions for the finite sublattices is NPcomplete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D nonplanar vs. 2D planar, (2) the nextnearest neighbour
Theory of Computation of Multidimensional Entropy with an Application to the MonomerDimer Problem
 Advances of Applied Math
"... We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multidimensional grid. As an application we compute the monomerdimer constant for the 2dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter ..."
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Cited by 20 (14 self)
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We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multidimensional grid. As an application we compute the monomerdimer constant for the 2dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3dimensional grid with an error smaller than 1.35%.
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 19 (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...
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 19 (10 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Matchings in Lattice Graphs
 Proceedings of the 25th ACM Symposium on Theory of Computing
, 1993
"... We study the problem of counting the number of matchings of given cardinality in a ddimensional rectangular lattice. This problem arises in several models in statistical physics, including monomerdimer systems and cellcluster theory. A classical algorithm due to Fisher, Kasteleyn and Temperley c ..."
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Cited by 7 (3 self)
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We study the problem of counting the number of matchings of given cardinality in a ddimensional rectangular lattice. This problem arises in several models in statistical physics, including monomerdimer systems and cellcluster theory. A classical algorithm due to Fisher, Kasteleyn and Temperley counts perfect matchings exactly in two dimensions, but is not applicable in higher dimensions and does not allow one to count matchings of arbitrary cardinality. In this paper, we present the first efficient approximation algorithms for counting matchings of arbitrary cardinality in (i) ddimensional "periodic" lattices (i.e., with wraparound edges) in any fixed dimension d; and (ii) twodimensional lattices with "fixed boundary conditions" (i.e., no wraparound edges). Our technique generalizes to approximately counting matchings in any bipartite graph that is the Cayley graph of some finite group. y CNRS, Ecole Normale Sup'erieure de Lyon, France. Part of this work was done while this au...
Bases Collapse in Holographic Algorithms
 Electronic Colloquium on Computational Complexity Report
, 2007
"... Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) c ..."
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Cited by 6 (1 self)
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Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #Pcomplete, and counting mod 2 is NPhard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2. 1