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 non-asymptotic 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.

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 ...

We discuss the problem of computing the volume of a convex body K in IR n . We review worst-case 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 CCR-8900112 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...

This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting prob-lems. 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.

We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, 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 NP-hard 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 NP-completeness in the early 1970's, the primary focus of complexity theory has been on decision ...

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...

We study the problem of learning a binary relation between two sets of objects or between a set and itself. We represent a binary relation between a set of size n and a set of size m as an n m matrix of bits, whose (i � j) entry is 1 if and only if the relation holds between the corresponding elements of the two sets. We present polynomial prediction algorithms for learning binary relations in an extended on-line 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. In the rst part of this paper, we present results for the case that the matrix of the relation has at most k row types. We present upper and lower bounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended on-line learning model. Furthermore, we describe a technique that simpli es the proof of expected mistake bounds against a randomly chosen query sequence. In the second part of this paper, we consider the problem of learning a binary relation that is a total order on a set. We describe a general technique using a fully

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 two-stage stochastic programming problems are ♯P-hard. Under the same assumption we show that certain multistage stochastic programming problems are PSPACE-hard. The problems we consider are non-standard in that distributions of stochastic parameters in later stages depend on decisions made in earlier stages.

Abstract not found

Abstract not found