A multipoint request is a group of collaborating nodes that wish to establish a communication for a certain duration of time. This need arises in parallel applications executed on processing elements connected either ...
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A multipoint request is a group of collaborating nodes that wish to establish a communication for a certain duration of time. This need arises in parallel applications executed on processing elements connected either by specialized interconnection networks or over wide area networks (collective communication operations). Each individual request is satisfied by a given subtree connecting the participating nodes. We aim to maximize the number of requests that can be simultaneously satisfied. In this paper, we show that this problem is NP-complete and we propose for it an approximation algorithm provided that the number of requests using the same edge is bounded by a constant.
The Task-Coalition Assignment Problem (TCAP) is a formalization of the distributed computation problem. In TCAP a set of agents and a set of tasks are given. A subset of the agents processes a task to produce benefit....
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The Task-Coalition Assignment Problem (TCAP) is a formalization of the distributed computation problem. In TCAP a set of agents and a set of tasks are given. A subset of the agents processes a task to produce benefit. The goal of TCAP is to find the combination of the tasks and the subsets of the agents that maximizes the sum of the benefit. In this paper, we define I-TCAP which is a practical subclass of TCAP. In 1-TCAP tasks and agents are characterized by scalar values. We propose a polynomial-time approximation algorithm for 1-TCAP and show that this algorithm achieves an approximation ratio 9/4. Here, an algorithm achieves an approximation ratio for a maximization problem if, for every instance, it produces a solution of value at least OPT/alpha, where OPT is the value of the optimal solution.
The constrained shortest path (CSP) problem requires the determination of a minimum costs-tpath with delay at most a nonzero *** this paper, we first point out the equivalence of certain algorithms, simply called the ...
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The constrained shortest path (CSP) problem requires the determination of a minimum costs-tpath with delay at most a nonzero *** this paper, we first point out the equivalence of certain algorithms, simply called the LARAC (Lagrangian Relaxation Based Aggregated Cost) algorithm presented independently in some earlier works. The LARAC algorithm solves the integer relaxation of the CSP problem (RELAX-CSP) and is based on a geometric approach. We then present an algebraic study of RELAX-CSP and establish several new properties of the optimal solution. These properties also hold for a class of combinatorial optimization problems involving two additive parameters. We follow this by establishing a characterization of optimal solutions for the general CSP problem involving more than two additive parameters. We present a new heuristic called LARAC-BIN based on binary search. This heuristic involves a parameter whose value can be specified in advance depending on the allowable deviation of the cost from the optimum. Using Megiddo's parametric search, we also present a strongly polynomial time algorithm for RELAX-CSP. This algorithm has the best complexity to date for RELAX-CSP. Finally, we present an integrated approach to the CSP problem and show how the LARAC algorithm can be used to achieve considerable speedup of ϵ-approximation algorithms for the CSP problem.
We study combinatorial optimization problems in which a set of distributed agents must achieve a global objective using only local information. Papadimitriou and Yannakakis [Proceedings of the 25th ACM Symposium on Th...
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We study combinatorial optimization problems in which a set of distributed agents must achieve a global objective using only local information. Papadimitriou and Yannakakis [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 121-129] initiated the study of such problems in a framework where distributed decision-makers must generate feasible solutions to positive linear programs with information only about local constraints. We extend their model by allowing these distributed decision-makers to perform local communication to acquire information over time and then explore the tradeoff between the amount of communication and the quality of the solution to the linear program that the decision-makers can obtain. Our main result is a distributed algorithm that obtains a (1 + epsilon) approximation to the optimal linear programming solution while using only a polylogarithmic number of rounds of local communication. This algorithm offers a significant improvement over the logarithmic approximation ratio previously obtained by Awerbuch and Azar [Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 240-249] for this problem while providing a comparable running time. Our results apply directly to the application of network flow control, an application in which distributed routers must quickly choose how to allocate bandwidth to connections using only local information to achieve global objectives. The sequential version of our algorithm is faster and considerably simpler than the best known approximation algorithms capable of achieving a (1 + epsilon) approximation ratio for positive linear programming.
The minimum a-fat decomposition problem is the problem of decomposing a simple polygon into fewest subpolygons, each with aspect ratio at most a, for a given alpha > 0. The main result in the paper is a polynomial ...
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The minimum a-fat decomposition problem is the problem of decomposing a simple polygon into fewest subpolygons, each with aspect ratio at most a, for a given alpha > 0. The main result in the paper is a polynomial time algorithm that solves the version of this problem that disallows Steiner points. The algorithm returns an optimal a-fat decomposition, if there is one, and reports failure otherwise. We also devise a faster approximation algorithm that produces, for any epsilon > 0, an (alpha + epsilon)-fat decomposition with as few polygons as an optimal a-fat decomposition. (C) 2004 Elsevier B.V All rights reserved.
The minimum a-fat decomposition problem is the problem of decomposing a simple polygon into fewest subpolygons, each with aspect ratio at most a, for a given alpha > 0. The main result in the paper is a polynomial ...
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The minimum a-fat decomposition problem is the problem of decomposing a simple polygon into fewest subpolygons, each with aspect ratio at most a, for a given alpha > 0. The main result in the paper is a polynomial time algorithm that solves the version of this problem that disallows Steiner points. The algorithm returns an optimal a-fat decomposition, if there is one, and reports failure otherwise. We also devise a faster approximation algorithm that produces, for any epsilon > 0, an (alpha + epsilon)-fat decomposition with as few polygons as an optimal a-fat decomposition. (C) 2004 Elsevier B.V All rights reserved.
The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of n positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized appro...
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The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of n positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized approximation algorithm for this problem with linear space complexity and time complexity of O(n log n). Experiments with random uniformly-distributed instances of SSP show that our algorithm outperforms, both in running time and average error, Martello and Toth's (1984) quadratic greedy search, whose time complexity is O(n2). We propose conjectures on the expected error of our algorithm for uniformly-distributed instances of SSP and provide some analytical arguments justifying these conjectures. We present also results of numerous tests. International Federation of Operational Research Societies 2002.
We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-Restricted MST Problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane...
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We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-Restricted MST Problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and epsilon > 0, the scheme finds an approximation with cost within 1 + epsilon of the lowest cost spanning tree with the property that all nodes have degree at most d. We also develop a polynomial time approximation scheme for the Euclidean version of the Red-Blue Separation Problem, again extending Arora's techniques. Given epsilon > 0, the scheme finds an approximation with cost within 1 + epsilon of the cost of the optimum separating polygon of the input nodes, in nearly linear time.
We consider makespan minimization for vehicle scheduling problems on trees with job requests that have release and handling times. 2-approximation algorithms were known for several variants of the single vehicle probl...
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We consider makespan minimization for vehicle scheduling problems on trees with job requests that have release and handling times. 2-approximation algorithms were known for several variants of the single vehicle problem on a path. A 3/2-approximation algorithm was known for the single vehicle problem on a path where there is a fixed starting point and the vehicle must return to the starting point upon completion. Karuno, Nagamochi and Ibaraki give a 2-approximation algorithm for the single vehicle problem on trees. We develop a polynomial time approximation scheme (PTAS) that runs in time linear in the number of job requests for the single vehicle scheduling problem on trees that have a constant number of leaves. This PTAS can be easily adapted to accommodate various starting/ending constraints. We then extended this to a PTAS for the multiple vehicle problem where vehicles operate in disjoint subtrees. (C) 2004 Elsevier B.V. All rights reserved.
This paper considers the problem of computing the squared volume of a largest j-dimensional simplex in an arbitrary d-dimensional polytope P given by its vertices (a "V-polytope"), for arbitrary integers j a...
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This paper considers the problem of computing the squared volume of a largest j-dimensional simplex in an arbitrary d-dimensional polytope P given by its vertices (a "V-polytope"), for arbitrary integers j and d with 1 less than or equal to j less than or equal to d. The problem was shown by Gritzmann, Klee and Larman to be NP-hard. This paper examines the possible accuracy of deterministic polynomial-time approximation algorithms for the problem. On the negative side, it is shown that unless P = NP, no such algorithm can approximately solve the problem within a factor of less than 1.09. It is also shown that the NP-hardness and inapproximability continue to hold when the polytope P is restricted to be an affine crosspolytope. On the positive side, a simple deterministic polynomial-time approximation algorithm for the problem is described. The algorithm takes as input integers j and d with i less than or equal to j less than or equal to d and a V-polytope P of dimension d. It returns a j-simplex S subset of P such that vol(2)(T)/vol(2)(S) less than or equal to A(Bj)(j), where T is any largest j-simplex in P, and A and B are positive constants independent of j, d, P. (C) 2003 Elsevier B.V. All rights reserved.
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