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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 316 (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 NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 196 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Hierarchical BOA Solves Ising Spin Glasses and MAXSAT
 In Proc. of the Genetic and Evolutionary Computation Conference (GECCO 2003), number 2724 in LNCS
, 2003
"... Theoretical and empirical evidence exists that the hierarchical Bayesian optimization algorithm (hBOA) can solve challenging hierarchical problems and anything easier. This paper applies hBOA to two important classes of realworld problems: Ising spinglass systems and maximum satis ability (MAX ..."
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Cited by 47 (18 self)
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Theoretical and empirical evidence exists that the hierarchical Bayesian optimization algorithm (hBOA) can solve challenging hierarchical problems and anything easier. This paper applies hBOA to two important classes of realworld problems: Ising spinglass systems and maximum satis ability (MAXSAT). The paper shows how easy it is to apply hBOA to realworld optimization problems. The results indicate that hBOA is capable of solving enormously dicult problems that cannot be solved by other optimizers and still provide competitive or better performance than problemspeci c approaches on other problems. The results thus con rm that hBOA is a practical, robust, and scalable technique for solving challenging realworld problems.
The SwendsenWang process does not always mix rapidly
 Proc. 29th ACM Symp. on Theory of Computing
, 1997
"... The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The l ..."
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Cited by 42 (3 self)
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The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the SwendsenWang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the SwendsenWang process requires exponential time (in the size of the system) to approach equilibrium.
The complexity of analog computation
 in Math. and Computers in Simulation 28(1986
"... We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP w ..."
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Cited by 39 (0 self)
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We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NPcomplete problem, 3SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by wellbehaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.
Optimization with extremal dynamics
 Physical Review Letters
"... We explore a new generalpurpose heuristic for finding highquality solutions to hard optimization problems. The method, called extremal optimization, is inspired by selforganized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successive ..."
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Cited by 36 (2 self)
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We explore a new generalpurpose heuristic for finding highquality solutions to hard optimization problems. The method, called extremal optimization, is inspired by selforganized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively replaces extremely undesirable variables of a single suboptimal solution with new, random ones. Large fluctuations ensue, that efficiently explore many local optima. With only one adjustable parameter, the heuristic’s performance has proven competitive with more elaborate methods, especially near phase transitions which are believed to coincide with the hardest instances. We use extremal optimization to elucidate the phase transition in the 3coloring problem, and we provide independent confirmation of previously reported extrapolations for the groundstate energy of±J spin glasses in d = 3 and 4. PACS number(s): 02.60.Pn, 05.65.+b, 75.10.Nr, 64.60.Cn. Many natural systems have, without any centralized organizing facility, developed into complex structures that optimize their use of resources in sophisticated ways [1]. Biological evolution has formed efficient
Quadratic forms on graphs
 Invent. Math
, 2005
"... We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R, ..."
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Cited by 35 (10 self)
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We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R,
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 30 (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
Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm
, 1995
"... In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 ..."
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Cited by 25 (3 self)
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In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 x 100 can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than 20 000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero field is 1.317.