The budgeted minimum cost flow problem with unit upgrading cost

Büsing, Christina; Koster, Arie M. C. A.; Kirchner, Sarah; Thome, Annika

doi:10.1002/net.21724

Cited by 8 publications

(8 citation statements)

References 17 publications

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“…We propose a pseudo‐polynomial algorithm for the multi‐budgeted matching problem with a fixed number of budget constraints on series‐parallel (SP) graphs and start with a formal definition of series‐parallel graphs following the one of Büsing et al .Definition Series‐parallel graphs can be recursively defined as follows: An edge

{s, t}

is an SP graph with distinguished source

s

and target

t

. Any graph that can be obtained by a finite number of the following two compositions of SP graphs is itself an SP graph. a)The series composition

G

of two SP graphs

G_{1}

with source

s_{1}

and target

t_{1}

and

G_{2}

with source

s_{2}

and target

t_{2}

is the graph obtained by contracting

t_{1}

and

s_{2}

.…”

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

“…The constructed mBM instance

I'

can therefore be solved in

O (| E (G^{T}) | \cdot \prod_{i = 1}^{k} B_{i}^{2})

time by Theorem 3.3. As

| E (G^{T}) | = | E (T) | + | A | \leq 2 | V (T) |

and the construction of

I'

can be done in

O (| V (T) |)

time the result follows. We note that such a transformation is in general not applicable for other budgeted problems, for example, the budgeted minimum cost flow problem .…”

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

“…We propose a pseudo-polynomial algorithm for the multi-budgeted matching problem with a fixed number of budget constraints on series-parallel (SP) graphs and start with a formal definition of series-parallel graphs following the one of Büsing et al [7]. Definition 3.1.…”

Section: Series-parallel Graphsmentioning

confidence: 99%

“…Theorem 3.6. The mBM with fixed k on trees can be solved in We note that such a transformation is in general not applicable for other budgeted problems, for example, the budgeted minimum cost flow problem [7].…”

Section: Lemma 35 Let T Be a Tree Then G T Is Series-parallelmentioning

confidence: 99%

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Multi‐budgeted matching problems

Büsing

Comis

2018

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The multi‐budgeted matching problem (mBM) is a weighted matching problem with k independent edge cost functions. For each cost function, a budget constraint requires the accumulated cost not to exceed a corresponding budget. We show that the mBM is strongly NP‐hard on paths with uniform edge weights and budgets by a reduction from 3‐SAT. Subsequently, we propose a dynamic program for series‐parallel graphs with pseudo‐polynomial run time for a fixed number of budget constraints. As an extension we show how this algorithm can be used to solve the mBM on trees using a graph transformation. Realizing that both these graph classes have a bounded treewidth in common, we introduce a dynamic program based on tree decompositions. This approach leads to a pseudo‐polynomial algorithm for the mBM with fixed k on graphs of bounded treewidth.

show abstract

{s, t}

is an SP graph with distinguished source

s

and target

t

. Any graph that can be obtained by a finite number of the following two compositions of SP graphs is itself an SP graph. a)The series composition

G

of two SP graphs

G_{1}

with source

s_{1}

and target

t_{1}

and

G_{2}

with source

s_{2}

and target

t_{2}

is the graph obtained by contracting

t_{1}

and

s_{2}

.…”

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

“…The constructed mBM instance

I'

can therefore be solved in

O (| E (G^{T}) | \cdot \prod_{i = 1}^{k} B_{i}^{2})

time by Theorem 3.3. As

| E (G^{T}) | = | E (T) | + | A | \leq 2 | V (T) |

and the construction of

I'

can be done in

O (| V (T) |)

time the result follows. We note that such a transformation is in general not applicable for other budgeted problems, for example, the budgeted minimum cost flow problem .…”

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

Section: Series-parallel Graphsmentioning

confidence: 99%

Section: Lemma 35 Let T Be a Tree Then G T Is Series-parallelmentioning

confidence: 99%

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Multi‐budgeted matching problems

Büsing

Comis

2018

Networks

Self Cite

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“…Büsing et al examine the budgeted minimum cost flow problem with unit upgrading costs; this extends the classical minimum cost flow problem by allowing one to reduce the cost of at most K arcs. The authors consider complexity and algorithms for the special case of an uncapacitated network with just one source.…”

mentioning

confidence: 99%

Preface: Static and dynamic optimization models for network routing problems

Gouveia¹,

Pióro²,

Rak³

2016

Networks

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On the complexity of the upgrading version of the maximal covering location problem

Baldomero‐Naranjo,

Kalcsics,

Rodríguez‐Chía

2024

Networks

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In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP‐hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in time for uniform weights and NP‐hard for non‐uniform weights. On paths, the single facility problem is solvable in time, while the ‐facility problem is NP‐hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo‐polynomial algorithm is developed for the single facility problem on trees with integer parameters.

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The budgeted minimum cost flow problem with unit upgrading cost

Cited by 8 publications

References 17 publications

Multi‐budgeted matching problems

Multi‐budgeted matching problems

Preface: Static and dynamic optimization models for network routing problems

On the complexity of the upgrading version of the maximal covering location problem

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