Diverse collections in matroids and graphs

We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems. The input to the Weighted Diverse Bases problem consists of a matroid M , a weight function ω : E ( M ) → N , and integers k ≥ 1 , d ≥ 1 . The task is to decide if there is a...

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Veröffentlicht in:Mathematical programming Jg. 204; H. 1-2; S. 415 - 447
Hauptverfasser: Fomin, Fedor V., Golovach, Petr A., Panolan, Fahad, Philip, Geevarghese, Saurabh, Saket
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Springer
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
68Q25
05B35
68W40
Diversity of solutions
05C85
68Q27
Matroids
Graphs
05C70
Parameterized complexity
ISSN:0025-5610, 1436-4646
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Zusammenfassung:We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems. The input to the Weighted Diverse Bases problem consists of a matroid M , a weight function ω : E ( M ) → N , and integers k ≥ 1 , d ≥ 1 . The task is to decide if there is a collection of k bases B 1 , ⋯ , B k of M such that the weight of the symmetric difference of any pair of these bases is at least d . The input to the Weighted Diverse Common Independent Sets problem consists of two matroids M 1 , M 2 defined on the same ground set E , a weight function ω : E → N , and integers k ≥ 1 , d ≥ 1 . The task is to decide if there is a collection of k common independent sets I 1 , ⋯ , I k of M 1 and M 2 such that the weight of the symmetric difference of any pair of these sets is at least d . The input to the Diverse Perfect Matchings problem consists of a graph G and integers k ≥ 1 , d ≥ 1 . The task is to decide if G contains k perfect matchings M 1 , ⋯ , M k such that the symmetric difference of any two of these matchings is at least d . We show that none of these problems can be solved in polynomial time unless P = NP . We derive fixed-parameter tractable ( FPT ) algorithms for all three problems with ( k , d ) as the parameter, and present a p o l y ( k , d ) -sized kernel for Weighted Diverse Bases .
Bibliographie:ObjectType-Article-1
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ObjectType-Feature-2
content type line 23
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01959-z