Results 1  10
of
23
Applying parallel computation algorithms in the design of serial algorithms
 J. ACM
, 1983
"... Abstract. The goal of this paper is to point out that analyses of parallelism in computational problems have practical implications even when multiprocessor machines are not available. This is true because, in many cases, a good parallel algorithm for one problem may turn out to be useful for design ..."
Abstract

Cited by 234 (7 self)
 Add to MetaCart
Abstract. The goal of this paper is to point out that analyses of parallelism in computational problems have practical implications even when multiprocessor machines are not available. This is true because, in many cases, a good parallel algorithm for one problem may turn out to be useful for designing an efficient serial algorithm for another problem. A d ~ eframework d for cases like this is presented. Particular cases, which are discussed in this paper, provide motivation for examining parallelism in sorting, selection, minimumspanningtree, shortest route, maxflow, and matrix multiplication problems, as well as in scheduling and locational problems.
Linear programming in linear time when the dimension is fixed
 J. ACM
, 1984
"... Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that i ..."
Abstract

Cited by 194 (13 self)
 Add to MetaCart
Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problemscomputations on matrices; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problemsgeometrical problems and computations; sorting and searching; G. 1.6 [Mathematics of Computing]: Optimizationlinear programming
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 & Co ..."
Abstract

Cited by 188 (0 self)
 Add to MetaCart
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
Tight Bounds and 2Approximation Algorithms for Integer Programs with Two Variables per Inequality
 Mathematical Programming
, 1992
"... . The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NPcomplete, even when all variables are binary. This paper deals with integer linear minimization problems in n variables subject to m linear constraints with at most tw ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
. The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NPcomplete, even when all variables are binary. This paper deals with integer linear minimization problems in n variables subject to m linear constraints with at most two variables per inequality, and with all variables bounded between 0 and U . For such systems, a 2\Gammaapproximation algorithm is presented that runs in time O(mnU 2 log(Un 2 =m)), so it is polynomial in the input size if the upper bound U is polynomially bounded. The algorithm works by finding first a superoptimal feasible solution that consists of integer multiples of 1 2 . That solution gives a tight bound on the value of the minimum. It further more has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, ...
Combinatorial Algorithms for the Generalized Circulation Problem
 MATHEMATICS OF OPERATIONS RESEARCH
, 1989
"... We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e)fl(e) units arrive at the other end. For instance, nodes of the graph can correspond to different curre ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e)fl(e) units arrive at the other end. For instance, nodes of the graph can correspond to different currencies, with the multipliers being the exchange rates. We require conservation of flow at every node except a given source node. The goal is to maximize the amount of flow excess at the source. This problem is a special case of linear programming, and therefore can be solved in polynomial time. In this paper we present the first polynomial time combinatorial algorithms for this problem. The algorithms are simple and intuitive.
Using Separation Algorithms in Fixed Dimension
 J. ALGORITHMS
, 1989
"... Consider a convex set in d dimensions. Assume that we are given a separation subroutine which, given a point, tells us whether this point is in the set. Moreover, if the point is not in the set, the subroutine separates the point from the set by a hyperplane. We show that if d is fixed and the se ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
Consider a convex set in d dimensions. Assume that we are given a separation subroutine which, given a point, tells us whether this point is in the set. Moreover, if the point is not in the set, the subroutine separates the point from the set by a hyperplane. We show that if d is fixed and the separation subroutine is linear in the input vector, this implies that one can optimize a linear objective function over the convex set in time polynomial in the number of arithmetic operations used by the separation subroutine. We apply this result to extend the class of linear programs solvable in strongly polynomial time. We show that a problem can be solved in strongly polynomial time if, by deleting a constant number of rows and columns, it can be converted to a problem which is already known to be solvable in strongly polynomial time. For example, this yields a strongly polynomial algorithm for the concurrent multicommodity flow problem.
Some Aspects Of The Combinatorial Theory Of Convex Polytopes
, 1993
"... . We start with a theorem of Perles on the kskeleton, Skel k (P ) (faces of dimension k) of d polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the kskeleton of a pyramid over a (d \Gamma 1)dimensional polytope. ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
. We start with a theorem of Perles on the kskeleton, Skel k (P ) (faces of dimension k) of d polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the kskeleton of a pyramid over a (d \Gamma 1)dimensional polytope. Therefore the number of combinatorially distinct kskeleta of dpolytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the gnumbers. For a dpolytope P there are [d=2] invariants g1 (P ); g2 (P ); :::; g [d=2] (P ) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems. Key words: Convex polytopes, skeleton, simplicial sphere, simplicial manifold, fvector, g theorem, ranked atomic lattices, stress, rigidity, sunflower, lower bound theorem, elementary poly...
A Polynomial Combinatorial Algorithm for Generalized Minimum Cost Flow
 In Proceedings of the 31th Annual ACM Symposium on Theory of Computing
, 1999
"... We propose the first combinatorial solution to one of the most classic problems in combinatorial optimization: the generalized minimum cost flow problem (flow with losses and gains). Despite a rich history dating back to Kantorovich and Dantzig, until now, the only known way to solve the problem ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
We propose the first combinatorial solution to one of the most classic problems in combinatorial optimization: the generalized minimum cost flow problem (flow with losses and gains). Despite a rich history dating back to Kantorovich and Dantzig, until now, the only known way to solve the problem in polynomialtime was via general purpose linear programming techniques. Polynomial combinatorial algorithms were previously known only for the version of our problem without costs. We design the first such algorithms for the version with costs. Our algorithms also find provably good solutions faster than optimal ones, providing the first strongly polynomial approximation schemes for the problem. Our techniques extend to optimize linear programs with two variables per inequality. Polynomial combinatorial algorithms were previously developed for testing the feasibility of such linear programs. Until now, no such methods were known for the optimization version. 1 Introduction In the ...
Complexity results for InfiniteHorizon Markov Decision Processes
, 2000
"... Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in a ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in artificial intelligence, has highlighted the significance of studying the computational properties of MDP problems. We address
Maximizing Concave Functions in Fixed Dimension
 in: Complexity in Numeric Computation
, 1993
"... In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading of this p ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading of this paper, we repeat some of the definitions and propositions given in [3, 5, 2]. Some proofs are not repeated, however, and instead we supply the interested reader with appropriate pointers. Suppose Q ae R d is a convex set given as an intersection of k halfspaces, and let g : Q ! R be a concave function that is computable by a piecewise affine algorithm (i.e., roughly, an algorithm that performs only multiplications by scalars, additions, and comparisons of intermediate values which depend on the input). Assume that such an algorithm A is given and the maximal number of operations required by A on any input (i.e., point in Q) is T . We show that under these assumptions, for any fixed d, the ...