A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007 ) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form  max { c T x : A x ≤ b , x ∈ C } , where the...

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Vydáno v:Mathematical methods of operations research (Heidelberg, Germany) Ročník 87; číslo 1; s. 19 - 50
Hlavní autoři: Holzhauser, Michael, Krumke, Sven O.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2018
Springer Nature B.V
Témata:
Algorithms
Approximation
Business and Management
Calculus of Variations and Optimal Control; Optimization
Cones
Functions (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations research
Operations Research/Decision Theory
Original Article
Packing problem
Topological manifolds
Fractional packing
Approximation algorithms
Network flows
ISSN:1432-2994, 1432-5217
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Popis
Shrnutí:We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007 ) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form  max { c T x : A x ≤ b , x ∈ C } , where the matrix  A contains no negative entries and C is a cone that is generated by a finite set  S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.
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ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-017-0604-2