Budget-constrained minimum cost flows

Holzhauser, Michael; Krumke, Sven O.; Thielen, Clemens

doi:10.1007/s10878-015-9865-y

Cited by 19 publications

(20 citation statements)

References 16 publications

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“…In particular, if Λ * denotes the set of all these values λ * , it holds that Λ * is a closed interval containing either one or infinitely many such values λ * depending on whether the optimum solutions correspond to points that lie amid or at the corner of some efficient edge in the objective space, respectively. As claimed in the following lemma, which is proven in [5], we are able to decide the membership in Λ * efficiently:…”

Section: Exact Algorithmsmentioning

confidence: 90%

“…Let MCF(m, n) denote the complexity of computing a traditional minimum-cost flow in strongly polynomial time. As shown in [5], Lemma 2 can be used within Megiddo's parametric search technique in order to obtain strongly polynomial-time algorithms for BCMCFP R :…”

Section: Exact Algorithmsmentioning

confidence: 99%

“…A related problem in which a fixed usage fee is induced by each edge with positive flow was investigated by Duque et al [4]. The model that we use here was recently investigated in Holzhauser et al [5], where a strongly polynomial-time algorithm based on the interpretation of the problem as a bicriteria minimum cost flow problem was derived.…”

Section: Introductionmentioning

confidence: 99%

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On the complexity and approximability of budget-constrained minimum cost flows

Holzhauser

Krumke

Thielen

2017

Information Processing Letters

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We investigate the complexity and approximability of the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total value in a feasible flow is constrained by a given budget B. This problem can, e.g., be seen as the application of the ε-constraint method to the bicriteria minimum cost flow problem. We show that we can solve the problem exactly in weakly polynomial time O(log M · MCF(m, n, C, U)), where C, U, and M are upper bounds on the largest absolute cost, largest capacity, and largest absolute value of any number occuring in the input, respectively, and MCF(m, n, C, U) denotes the complexity of finding a traditional minimum cost flow. Moreover, we present two fully polynomial-time approximation schemes for the problem on general graphs and one with an improved running-time for the problem on acyclic graphs.

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Section: Exact Algorithmsmentioning

confidence: 90%

Section: Exact Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the complexity and approximability of budget-constrained minimum cost flows

Holzhauser

Krumke

Thielen

2017

Information Processing Letters

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“…In the budget-constrained minimum cost flow problem, the aim is to determine a minimum cost flow subject to a budget constraint of the form e∈E b e · x e B, similarly to the budget-constrained maximum flow problem that was studied above. The problem is known to be efficiently solvable in weakly and strongly polynomial-time [22,23]. In [22], a strongly polynomial-time FPTAS was presented for the budget-constrained minimum cost flow problem, which runs in O 1 ε 2 · m log m · (nm log m log log m + n 3 log n + nm log 2 n log log n) time and which uses similar ideas as the ones presented above.…”

Section: Budget-constrained Minimum Cost Flowsmentioning

confidence: 99%

A generalized approximation framework for fractional network flow and packing problems

Holzhauser

Krumke

2017

Math Meth Oper Res

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We generalize the fractional packing framework of Garg and Koenemann [17] 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 : Ax 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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“…Holzhauser et al describe a MCF problem with the extension of a second cost per arc. This second cost applies as soon as there is positive flow on the arc.…”

Section: Introductionmentioning

confidence: 99%

The budgeted minimum cost flow problem with unit upgrading cost

et al. 2016

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The budgeted minimum cost flow problem (BMCF(K)) with unit upgrading costs extends the classical minimum cost flow problem by allowing one to reduce the cost of at most K arcs. In this article, we consider complexity and algorithms for the special case of an uncapacitated network with just one source. By a reduction from 3‐SAT we prove strong N P ‐completeness and inapproximability, even on directed acyclic graphs. On the positive side, we identify three polynomially solvable cases: on arborescences, on so‐called tree‐like graphs, and on instances with a constant number of sinks. Furthermore, we develop dynamic programs with pseudo‐polynomial running time for the BMCF(K) problem on (directed) series‐parallel graphs and (directed) graphs of bounded treewidth. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 67–82 2017

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Budget-constrained minimum cost flows

Cited by 19 publications

References 16 publications

On the complexity and approximability of budget-constrained minimum cost flows

On the complexity and approximability of budget-constrained minimum cost flows

A generalized approximation framework for fractional network flow and packing problems

The budgeted minimum cost flow problem with unit upgrading cost

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