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78
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 286 (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.
Counting Solutions to Linear and Nonlinear Constraints through Ehrhart Polynomials: Applications to Analyze and Transform Scientific Programs
, 1996
"... In order to produce efficient parallel programs, optimizing compilers need to include an analysis of the initial sequential code. When analyzing loops with affine loop bounds, many computations are relevant to the same general problem: counting the number of integer solutions of selected free variab ..."
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Cited by 107 (0 self)
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In order to produce efficient parallel programs, optimizing compilers need to include an analysis of the initial sequential code. When analyzing loops with affine loop bounds, many computations are relevant to the same general problem: counting the number of integer solutions of selected free variables in a set of linear and/or nonlinear parameterized constraints. For example, computing the number of flops executed by a loop, of memory locations touched by a loop, of cache lines touched by a loop, or of array elements that need to be transmitted from a processor to another during the execution of a loop, is useful to determine if a loop is load balanced, evaluate message traffic and allocate message buffers. The objective of the presented method is to evaluate symbolically, in terms of symbolic constants (the size parameters) , this number of integer solutions. By modeling the considered counting problem as a union of rational convex polytopes, the number of included integer points is ...
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 90 (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 ...
Computing The Volume Of Convex Bodies: A Case Where Randomness Provably Helps
, 1991
"... We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide som ..."
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Cited by 78 (7 self)
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We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide some applications of this latter result. Supported by NATO grant RG0088/89 y Supported by NSF grant CCR8900112 and NATO grant RG0088/89 1 Introduction The mathematical study of areas and volumes is as old as civilization itself, and has been conducted for both intellectual and practical reasons. As far back as 2000 B.C., the Egyptians 1 had methods for approximating the areas of fields (for taxation purposes) and the volumes of granaries. The exact study of areas and volumes began with Euclid 2 and was carried to a high art form by Archimedes 3 . The modern study of this subject began with the great astronomer Johann Kepler's treatise 4 Nova stereometria doliorum vinariorum, wh...
Computational complexity of stochastic programming problems
, 2005
"... Stochastic programming is the subfield of mathematical programming that considers optimization in the presence of uncertainty. During the last four decades a vast quantity of literature on the subject has appeared. Developments in the theory of computational complexity allow us to establish the theo ..."
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Cited by 57 (1 self)
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Stochastic programming is the subfield of mathematical programming that considers optimization in the presence of uncertainty. During the last four decades a vast quantity of literature on the subject has appeared. Developments in the theory of computational complexity allow us to establish the theoretical complexity of a variety of stochastic programming problems studied in this literature. Under the assumption that the stochastic parameters are independently distributed, we show that twostage stochastic programming problems are ♯Phard. Under the same assumption we show that certain multistage stochastic programming problems are PSPACEhard. The problems we consider are nonstandard in that distributions of stochastic parameters in later stages depend on decisions made in earlier stages.
Sparse Elimination and Applications in Kinematics
 UC BERKELEY
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 52 (11 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
Markov Chains and Polynomial time Algorithms
, 1994
"... This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing ..."
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Cited by 44 (0 self)
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This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.
LEARNING BINARY RELATIONS AND TOTAL ORDERS
, 1993
"... The problem of learning a binary relation between two sets of objects or between a set and itself is studied. This paper represents a binary relation between a set of size n and a set of size rn as an n rn matrix of bits whose (i, j) entry is if and only if the relation holds between the correspond ..."
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Cited by 36 (5 self)
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The problem of learning a binary relation between two sets of objects or between a set and itself is studied. This paper represents a binary relation between a set of size n and a set of size rn as an n rn matrix of bits whose (i, j) entry is if and only if the relation holds between the corresponding elements of the two sets. Polynomial prediction algorithms are presented for learning binary relations in an extended online learning model, where the examples are drawn by the learner, by a helpful teacher, by an adversary, or according to a uniform probability distribution on the instance space. The first part of this paper presents results for the case in which the matrix of the relation has at most k row types. It presents upper and lower bounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended online learning model. Furthermore, it describes a technique that simplifies the proof of expected mistake bounds against a randomly chosen query sequence. In the second part of this paper the problem of learning a binary relation that is a total order on a set is considered. A general technique using a fully polynomial randomized approximation scheme (fpras) to implement a randomized version of the halving algorithm is described. This technique is applied to the problem of learning a total order, through the use of an fpras for counting the number of extensions of a partial order, to obtain a polynomial prediction algorithm that with high probability makes at most n lg n + (lg e)lg n mistakes when an adversary selects the query sequence. The case in which a teacher or the learner selects the query sequence is also considered
Approximating the volume of unions and intersections of highdimensional geometric objects
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Exact volume computation for polytopes: a practical study
 in: Polytopes{combinatorics and computation (Oberwolfach
, 1997
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