Budgeted Colored Matching Problems

Büsing, Christina; Comis, Martin

doi:10.1016/j.endm.2018.01.026

Cited by 2 publications

(7 citation statements)

References 9 publications

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“…Since the multi-budgeted matching problem generalizes the budgeted colored matching problem, its strong NP-hardness on bipartite graphs with uniform edge weights and budgets follows immediately from the corresponding strong NP-hardness result of the latter by Büsing and Comis [6]. We strengthen this result by showing that even on paths with uniform edge weights and budgets the mBM is strongly NP-hard.…”

Section: O M P L E X I T Ysupporting

confidence: 58%

“…Until now this conjecture could be neither proved, nor disproved. Both the reduction presented in this article as well as the reduction for the BCM in rely on variants of the 3‐SAT problem and require one budget constraint per clause. However, for a fixed number of clauses 3‐SAT is polynomial‐time solvable by simple enumeration and thus every attempt at proving strong NP‐hardness has to be based on a new construction.…”

Section: Resultsmentioning

confidence: 99%

“…Therefore, we focus on the multi‐budgeted matching problem with a fixed number of budget constraints. This section is in content and structure closely related to the respective section in Büsing and Comis and the presented dynamic programs are generalizations of the dynamic programs for the BCM presented therein.…”

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

“…A budgeted matching problem on edge‐colored graphs studied by Büsing and Comis is the budgeted colored matching problem (BCM). In the BCM we have a single cost function and one budget constraint for every color in the edge‐coloring, limiting the total cost of edges in the respective color.…”

Section: Introductionmentioning

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.

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Section: O M P L E X I T Ysupporting

confidence: 58%

Section: Resultsmentioning

confidence: 99%

Section: Series‐parallel Graphs and Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multi‐budgeted matching problems

Büsing

Comis

2018

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“…Moreover, several bi-criteria approximation algorithms for BCM, which are allowed to slightly violate the color constraints, are due to Mastrolilli and Stamoulis [21,22]. Recently, an extension of BCM that additionally incorporates edge costs was studied under the name budgeted colored matching problem [7]. Büsing and Comis [7] present pseudo-polynomial dynamic programs for the budgeted colored matching problem with a fixed number of colors on series-parallel graphs and trees.…”

Section: Introductionmentioning

confidence: 99%

Minimum color‐degree perfect b‐matchings

et al. 2020

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The minimum color‐degree perfect b‐matching problem (Col‐BM) is a new extension of the perfect b‐matching problem to edge‐colored graphs. The objective of Col‐BM is to minimize the maximum number of differently colored edges in a perfect b‐matching that are incident to the same node. We show that Col‐BM is NP‐hard on bipartite graphs by a reduction from (3,B2)‐Sat, and conclude that there exists no (2 − ϵ)‐approximation algorithm unless P=NP. However, we identify a class of two‐colored complete bipartite graphs on which we can solve Col‐BM in polynomial time. Furthermore, we use dynamic programming to devise polynomial‐time algorithms solving Col‐BM with a fixed number of colors on series‐parallel graphs and simple graphs with bounded treewidth.

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Budgeted Colored Matching Problems

Cited by 2 publications

References 9 publications

Multi‐budgeted matching problems

Multi‐budgeted matching problems

Minimum color‐degree perfect b‐matchings

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