Minimum-cost integer circulations in given homology classes

Morell, Sarah; Seidel, Ina; Weltge, Stefan

doi:10.1137/1.9781611976465.162

Cited by 4 publications

(3 citation statements)

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“…The main result of [4] (see also Morell, Seidel and Weltge [31]) tells us how to solve the problem on the surface part G 0 alone. If we knew that the feasible solution avoids the boundary cycle of each G i , i ∈ [t], then we would be done after performing a single homologous circulation computation on G * 0 , and taking the empty solution within each G i , i ∈ [t].…”

Section: Overviewmentioning

confidence: 99%

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Integer programs with bounded subdeterminants and two nonzeros per row

Fiorini¹,

Joret²,

Weltge³

et al. 2021

Preprint

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We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k = 0 (bipartite graphs) and for k = 1.We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to b-matching.

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Section: Overviewmentioning

confidence: 99%

“…φ(e) = 0 for all other edges incident to a vortex face, and ω(φ) = ψ. These values can be precomputed in strongly polynomial time, see [4] or Morell, Seidel and Weltge [31].…”

Section: Integer Programs With Two Nonzero Entries Per Columnmentioning

confidence: 99%

Integer programs with bounded subdeterminants and two nonzeros per row

Fiorini¹,

Joret²,

Weltge³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…A closely related problem is the problem of computing a max-flow in a graph which admits an embedding into some topological space. The most well-studied cases are planar graphs and the more general case when the graph embeds into a surface [3,4,5,12,14,15,16,20,21,24,25]. Max-flows and min-cuts are computationally easier to solve in surface embedded graphs, especially planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Generalized max-flows and min-cuts in simplicial complexes

Maxwell,

Nayyeri

2021

Preprint

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We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For d-dimensional simplicial complexes embedded into R d+1 we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.

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