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An FPTAS for optimizing a class of lowrank functions over a polytope
, 2011
"... We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Paretooptimal front of the linear functions which constitute the given lowrank func ..."
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Cited by 7 (1 self)
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We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Paretooptimal front of the linear functions which constitute the given lowrank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasiconcavity on the objective function. For the special case of quasiconcave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with nonlinear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.
A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One
"... Abstract. In this paper, we propose a general framework for designing fully polynomial time approximation schemes for combinatorial optimization problems, in which more than one objective function are combined into one using any norm. The main idea is to exploit the approximate Paretooptimal fronti ..."
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Abstract. In this paper, we propose a general framework for designing fully polynomial time approximation schemes for combinatorial optimization problems, in which more than one objective function are combined into one using any norm. The main idea is to exploit the approximate Paretooptimal frontier for multicriteria optimization problems. Using this approach, we obtain an FPTAS for a novel resource allocation problem, for the problem of scheduling jobs on unrelated parallel machines, and for the Santa Claus problem, when the number of agents/machines is fixed, for any norm, including the l∞norm. Moreover, either FPTAS can be implemented in a manner so that the space requirements are polynomial in all input parameters. We also give approximation algorithms and hardness results for the resource allocation problem when the number of agents is not fixed. 1
Bilinear games: Polynomial time algorithms for rank based subclasses
 IN WORKSHOP IN INTERNET AND NETWORK ECONOMICS
, 2011
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An FPTAS for Minimizing a Class of LowRank QuasiConcave Functions over a Convex Set
"... We consider minimizing a class of low rank quasiconcave functions over a convex set and give a fully polynomial time approximation scheme (FPTAS) for the problem. The algorithm is based on a binary search for the optimal objective value which is guided by solving a polynomial number of linear minim ..."
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We consider minimizing a class of low rank quasiconcave functions over a convex set and give a fully polynomial time approximation scheme (FPTAS) for the problem. The algorithm is based on a binary search for the optimal objective value which is guided by solving a polynomial number of linear minimization problems over the convex set with appropriate objective functions. Our algorithm gives a (1 + )approximate solution that is an extreme point of the convex set and therefore, has direct applications to combinatorial 01 problems for which the convex hull of feasible solutions is known, such as shortest paths, spanning trees and matchings in undirected graphs. Our results also extend to maximization of lowrank quasiconvex functions over a convex set.
Algorithms for Discrete, NonLinear and Robust Optimization Problems with Applications in Scheduling and Service Operations
, 2011
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