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97
Planning and acting in partially observable stochastic domains
 ARTIFICIAL INTELLIGENCE
, 1998
"... In this paper, we bring techniques from operations research to bear on the problem of choosing optimal actions in partially observable stochastic domains. We begin by introducing the theory of Markov decision processes (mdps) and partially observable mdps (pomdps). We then outline a novel algorithm ..."
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Cited by 822 (31 self)
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In this paper, we bring techniques from operations research to bear on the problem of choosing optimal actions in partially observable stochastic domains. We begin by introducing the theory of Markov decision processes (mdps) and partially observable mdps (pomdps). We then outline a novel algorithm for solving pomdps offline and show how, in some cases, a finitememory controller can be extracted from the solution to a pomdp. We conclude with a discussion of how our approach relates to previous work, the complexity of finding exact solutions to pomdps, and of some possibilities for finding approximate solutions.
Interval propagation to reason about sets: definition and implementation of a practical language
 CONSTRAINTS
, 1997
"... Local consistency techniques have been introduced in logic programming in order to extend the application domain of logic programming languages. The existing languages based on these techniques consider arithmetic constraints applied to variables ranging over nite integer domains. This makes difficu ..."
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Cited by 102 (5 self)
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Local consistency techniques have been introduced in logic programming in order to extend the application domain of logic programming languages. The existing languages based on these techniques consider arithmetic constraints applied to variables ranging over nite integer domains. This makes difficult a natural and concise modelling as well as an efficient solving of a class of NPcomplete combinatorial search problems dealing with sets. To overcome these problems, we propose a solution which consists in extending the notion of integer domains to that of set domains (sets of sets). We specify a set domain by an interval whose lower and upper bounds are known sets, ordered by set inclusion. We define the formal and practical framework of a new constraint logic programming language over set domains, called Conjunto. Conjunto comprises the usual set operation symbols ([ � \ � n), and the set inclusion relation (). Set expressions built using the operation symbols are interpreted as relations (s [ s1 = s2,...). In addition, Conjunto provides us with a set of constraints called graduated constraints (e.g. the set cardinality) which map sets onto arithmetic terms. This allows us to handle optimization problems by applying a cost function to the quantifiable, i.e., arithmetic, terms which are associated to set terms. The constraint solving in Conjunto is based on local consistency techniques using interval reasoning which are extended to handle set constraints. The main contribution of this paper concerns the formal definition of the language and its design and implementation as a practical language.
Loop Parallelization in the Polytope Model
 CONCUR '93, Lecture Notes in Computer Science 715
, 1993
"... . During the course of the last decade, a mathematical model for the parallelization of FORloops has become increasingly popular. In this model, a (perfect) nest of r FORloops is represented by a convex polytope in Z r . The boundaries of each loop specify the extent of the polytope in a dis ..."
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Cited by 94 (23 self)
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. During the course of the last decade, a mathematical model for the parallelization of FORloops has become increasingly popular. In this model, a (perfect) nest of r FORloops is represented by a convex polytope in Z r . The boundaries of each loop specify the extent of the polytope in a distinct dimension. Various ways of slicing and segmenting the polytope yield a multitude of guaranteed correct mappings of the loops' operations in spacetime. These transformations have a very intuitive interpretation and can be easily quantified and automated due to their mathematical foundation in linear programming and linear algebra. With the recent availability of massively parallel computers, the idea of loop parallelization is gaining significance, since it promises execution speedups of orders of magnitude. The polytope model for loop parallelization has its origin in systolic design, but it applies in more general settings and methods based on it will become a part of futur...
A Geometric Buchberger Algorithm for Integer Programming
 Mathematics of Operations Research
, 1995
"... Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution α to a program in this family, there is at lea ..."
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Cited by 56 (10 self)
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Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution α to a program in this family, there is at least one element g in this set such that α  g has an improved cost value as compared to α. We describe a unique minimal test set for this family called the reduced Gröbner basis of IP. An algorithm for its construction is presented which we call a Geometric Buchberger Algorithm for integer programming. We show how an integer program may be solved using this test set and examine some geometric properties of elements in the set. The reduced Grobner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A ∈ Z^(n2)×n of full row rank.
Optimizing Data Locality by Array Restructuring
, 1995
"... It is increasingly important that optimizing compilers restructure programs for data locality to obtain high performance on today's powerful architectures. In this paper, we focus on array restructuring , a technique that improves the spatial locality exhibited by array accesses in nested loops. Spe ..."
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Cited by 43 (0 self)
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It is increasingly important that optimizing compilers restructure programs for data locality to obtain high performance on today's powerful architectures. In this paper, we focus on array restructuring , a technique that improves the spatial locality exhibited by array accesses in nested loops. Specifically, we address the following question: Given a set of such accesses, how should the array elements be laid out in memory to match the access pattern and thus maximize locality? Our approach is based on an invertible linear transformation of array index vectors. We present algorithms to choose a suitable transformation, and hence array layout, given the set of array accesses. Our analysis places no restrictions on the loop's nesting structure or dependence pattern. Although we focus on cases where the array indexing expressions are affine functions of loop variables, our techniques can be applied to the nonaffine case as well. We have implemented our technique in the SUIF compiler [17...
Variation of Cost Functions in Integer Programming
 MATHEMATICAL PROGRAMMING
, 1994
"... We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classe ..."
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Cited by 42 (8 self)
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We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Gröbner bases associated with A. The union of the reduced Gröbner bases as c varies (called the universal Gröbner basis) consists precisely of the edge directions of St(A). We present geometric algorithms for computing St(A), the Graver basis [Gra], and the universal Gröbner basis.
Short rational generating functions for lattice point problems
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2003
"... We prove that for any fixed d the generating function of the projection of the set of integer points in a rational ddimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert ..."
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Cited by 40 (5 self)
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We prove that for any fixed d the generating function of the projection of the set of integer points in a rational ddimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a nonnegative integer combination of given coprime positive integers a1,..., ad) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.
Foundations of Aggregation Constraints
, 1994
"... We introduce a new constraint domain, aggregation constraints, that is useful in database query languages, and in constraint logic programming languages that incorporate aggregate functions. We formally study the fundamental problem of determining if a conjunction of aggregation constraints is satis ..."
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Cited by 38 (1 self)
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We introduce a new constraint domain, aggregation constraints, that is useful in database query languages, and in constraint logic programming languages that incorporate aggregate functions. We formally study the fundamental problem of determining if a conjunction of aggregation constraints is satisfiable, and show that, for many classes of aggregation constraints, the problem is undecidable. We describe a complete and minimal axiomatization of aggregation constraints, for the SQL aggregate functions min, max, sum, count and average, over a nonempty, finite multiset on several domains. This axiomatization helps identify classes of aggregation constraints for which the satisfiability check is efficient. We present a polynomialtime algorithm that directly checks for satisfiability of a conjunction of aggregation range constraints over a single multiset; this is a practically useful class of aggregation constraints. We discuss the relationships between aggregation constraints over a non...
CommunicationMinimal Tiling of Uniform Dependence Loops
, 1996
"... . Tiling is a loop transformation that a compiler uses to create automatically blocked algorithms in order to improve the benefits of the memory hierarchy and reduce the communication overhead between processors. Motivated by existing results, this paper presents a conceptually simple approach to fi ..."
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Cited by 37 (4 self)
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. Tiling is a loop transformation that a compiler uses to create automatically blocked algorithms in order to improve the benefits of the memory hierarchy and reduce the communication overhead between processors. Motivated by existing results, this paper presents a conceptually simple approach to finding tilings with a minimal amount of communication between tiles. The development of almost all results is based primarily on the inequality of arithmetic and geometric means and the concept of extremal rays from convex cones. The key insight is that a tiling that is communicationminimal must induce the same amount of communication through all faces of a tile, which restricts the search space for optimal tilings to those tiling matrices whose rows are all extremal rays in a cone. For nested loops with several special forms of dependences, closedform optimal tilings are derived. In the general case, a procedure is given that always returns optimal tilings. An efficient implementation of t...
An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems
, 1993
"... are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares ..."
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Cited by 36 (4 self)
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are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares penalty function. The problem of characterizing and identifying quadratic maps with convex images is analyzed from both structural and complexity theoretic points of view. Then a study is made of the geometry of a class of convex sets called spectrahedra, which are the feasible regions in semide nite programs. Finally, in Chapter 7, we develop some cutting plane techniques for MQP, based on eigenvalue inequalities. Acknowledgements I express my sincere gratitude to my thesis advisor Professor Alan Goldman for his support, ideas and encouragement. My special thanks to Professors Laszlo Lovasz and James Renegar for sparing their time generously and giving me very useful suggestions. I thank the warm and friendly Professors Dan Naiman and Ed Scheinerman for making my four year long stay at Johns Hopkins a very pleasant one. I am indebted to Prof. JongShi Pang and Prof. Roger Horn for giving a patient ear to many of my enthusiastic ideas and o ering suggestions. I also thank Prof. ShihPing Han for being a wonderful teacher, and Prof. Leslie Hall for being a patient second reader of my thesis. My interest in Multiquadratic Programming was initiated during my internship at AT&T Bell Laboratories in the summer of 1990, and I am obliged to Dr. Narendra Karmarkar for arranging this internship. I thank Dr. Farid Alizadeh, Dr. Florian Jarre, Profs. Raphael Loewy, Michael Overton and Stephen Vavasis for patiently answering my questions and sending me some literature. My heartfelt appreciation is due my parents, Lakshmi Pathy and Satyavathi, my sister,