Minimum cost noncrossing flow problem on layered networks

Altınel, İ. Kuban; Aras, Necatı; Şuvak, Zeynep; Taşkın, Z. Caner

doi:10.1016/j.dam.2018.09.016

Cited by 7 publications

(10 citation statements)

References 18 publications

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“…Their problem is to find the noncrossing flow with minimum total cost, which is called as the minimum cost noncrossing flow problem (MCNFP). In this work, we extend the previous results by [2] for general flow networks and disjunctive conflicts where neither the underlying network nor the conflicting arcs have a special structure. To the best of our knowledge, this work is the first study of the conflict variant of the minimum cost flow problem on general networks.…”

Section: Introductionsupporting

confidence: 67%

“…It would not be surprising that MCFPC is NP ‐hard given the intractability results for the similar problems TPC [25], MFPC [39], and MCNFP [2]. Nevertheless, we provide a more formal complexity analysis below.…”

Section: Complexity and Formulationmentioning

confidence: 99%

“…One of them is the transportation problem with conflicts (TPC) which is studied by [7, 8, 25, 47, 48, 50]. The maximum flow problem with conflict constraints (MFPC) studied by [40], and the minimum cost noncrossing flow problem (MCNFP) addressed in [2] are other more recent examples.…”

Section: Introductionmentioning

confidence: 99%

“…Goossens and Spieksma [25] analyze the difficulty of TPC; they show that even some particular classes with identical exclusionary sets, a single exclusionary set, and a fixed number of supply nodes, are NP ‐complete. The works by [2, 40, 49] are more recent. The first one is theoretical and includes complexity results on MFPC and its relatives.…”

Section: Introductionmentioning

confidence: 99%

“…The problem of our concern, namely MCFPC, is first introduced in [2] within a more specific context for layered networks. They analyze the complexity of this problem for the situation where two arcs are in conflict only if they physically cross each other with respect to a particular embedding of vertices within a layer.…”

Section: Introductionmentioning

confidence: 99%

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Minimum cost flow problem with conflicts

2021

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The minimum cost flow problem with conflicts is a recent extension of the ordinary minimum cost flow problem. It includes flow compatibility restrictions in addition to flow balance equalities and capacity constraints: at most one of the conflicting arcs can have positive flow. In this work we study this extension of the minimum cost flow problem, which can be faced in many real‐world applications. We give mixed‐integer programming formulations of the problem after briefly analyzing its complexity and propose two exact solution algorithms. One of them is a branch‐and‐bound procedure with combinatorial branching rules based on conflicts in an optimal solution of the relaxations. The other one is a Russian Doll Search method exploring the maximal stable sets of the conflict graph, which is obtained from the original network with the purpose of representing the conflict relations among arcs. Also, a set of preprocessing procedures is introduced with the aim of reducing the size of the problem. We can say that the new algorithms are very efficient according to the results of extensive computational experiments realized on a large set of test problems.

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Section: Introductionsupporting

confidence: 67%

Section: Complexity and Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Minimum cost flow problem with conflicts

2021

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Maximum weight perfect matching problem with additional disjunctive conflict constraints

2022

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We focus on an extension of the maximum weight perfect matching problem with additional disjunctive conflict constraints in conjunction with the degree and binary restrictions. Given a simple graph with a nonnegative weight associated with each edge and a set of conflicting edges, the perfect matching problem with conflict constraints consists of finding a maximum weight perfect matching without any conflicting edge pair. Unlike the well‐known ordinary maximum weight perfect matching problem this one is strongly 𝒩𝒫‐hard. We propose two branch‐and‐bound algorithms for the exact solution of the problem. The first one is based on an equivalent maximum weight stable set formulation with an additional cardinality restriction obtained on the graph representing conflict relations and uses the information coming from its maximal stable sets. The second one is essentially a recursive depth first search scheme that benefits from simple upper bounds incorporated with a fast infeasibility detection procedure to prune the branch‐and‐bound tree. According to the extensive computational tests it is possible to say that they are both very efficient.

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