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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Published in:	Mathematical programming Vol. 204; no. 1-2; pp. 415 - 447
Main Authors:	Fomin, Fedor V., Golovach, Petr A., Panolan, Fahad, Philip, Geevarghese, Saurabh, Saket
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024 Springer
Subjects:	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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Abstract	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 .
AbstractList	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 [Formula omitted], a weight function [Formula omitted], and integers [Formula omitted]. The task is to decide if there is a collection of [Formula omitted]bases [Formula omitted] of [Formula omitted] such that the weight of the symmetric difference of any pair of these bases is at least [Formula omitted]. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids [Formula omitted] defined on the same ground set [Formula omitted], a weight function [Formula omitted], and integers [Formula omitted]. The task is to decide if there is a collection of [Formula omitted]common independent sets [Formula omitted] of [Formula omitted] and [Formula omitted] such that the weight of the symmetric difference of any pair of these sets is at least [Formula omitted]. The input to the Diverse Perfect Matchings problem consists of a graph [Formula omitted] and integers [Formula omitted]. The task is to decide if [Formula omitted] contains [Formula omitted]perfect matchings [Formula omitted] such that the symmetric difference of any two of these matchings is at least [Formula omitted]. We show that none of these problems can be solved in polynomial time unless [Formula omitted]. We derive fixed-parameter tractable ( [Formula omitted]) algorithms for all three problems with [Formula omitted] as the parameter, and present a [Formula omitted]-sized kernel for Weighted Diverse Bases. 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 , a weight function , and integers . The task is to decide if there is a collection of of such that the weight of the symmetric difference of any pair of these bases is at least . The input to the Weighted Diverse Common Independent Sets problem consists of two matroids defined on the same ground set , a weight function , and integers . The task is to decide if there is a collection of of and such that the weight of the symmetric difference of any pair of these sets is at least . The input to the Diverse Perfect Matchings problem consists of a graph and integers . The task is to decide if contains such that the symmetric difference of any two of these matchings is at least . We show that none of these problems can be solved in polynomial time unless . We derive fixed-parameter tractable ( ) algorithms for all three problems with as the parameter, and present a -sized kernel for Weighted Diverse Bases. 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 $$M$$ M , a weight function $$\omega :E(M)\rightarrow \mathbb {N} $$ ω : E ( M ) → N , and integers $$k\ge 1, d\ge 1$$ k ≥ 1 , d ≥ 1 . The task is to decide if there is a collection of $$k$$ k bases $$B_{1}, \dotsc , B_{k}$$ B 1 , ⋯ , B k of $$M$$ M such that the weight of the symmetric difference of any pair of these bases is at least $$d$$ d . The input to the Weighted Diverse Common Independent Sets problem consists of two matroids $$M_{1},M_{2}$$ M 1 , M 2 defined on the same ground set $$E$$ E , a weight function $$\omega :E\rightarrow \mathbb {N} $$ ω : E → N , and integers $$k\ge 1, d\ge 1$$ k ≥ 1 , d ≥ 1 . The task is to decide if there is a collection of $$k$$ k common independent sets $$I_{1}, \dotsc , I_{k}$$ I 1 , ⋯ , I k of $$M_{1}$$ M 1 and $$M_{2}$$ M 2 such that the weight of the symmetric difference of any pair of these sets is at least $$d$$ d . The input to the Diverse Perfect Matchings problem consists of a graph $$G$$ G and integers $$k\ge 1, d\ge 1$$ k ≥ 1 , d ≥ 1 . The task is to decide if $$G$$ G contains $$k$$ k perfect matchings $$M_{1},\dotsc ,M_{k}$$ M 1 , ⋯ , M k such that the symmetric difference of any two of these matchings is at least $$d$$ d . We show that none of these problems can be solved in polynomial time unless $${{\,\mathrm{\textsf{P}}\,}} ={{\,\mathrm{\textsf{NP}}\,}} $$ P = NP . We derive fixed-parameter tractable ( $${{\,\mathrm{\textsf{FPT}}\,}}$$ FPT ) algorithms for all three problems with $$(k,d)$$ ( k , d ) as the parameter, and present a $$poly(k,d)$$ p o l y ( k , d ) -sized kernel for Weighted Diverse Bases . 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 . 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 B1,⋯,Bk 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 M1,M2 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 I1,⋯,Ik of M1 and M2 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 M1,⋯,Mk 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 poly(k,d)-sized kernel for Weighted Diverse Bases.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 B1,⋯,Bk 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 M1,M2 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 I1,⋯,Ik of M1 and M2 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 M1,⋯,Mk 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 poly(k,d)-sized kernel for Weighted Diverse Bases.
Audience	Academic
Author	Fomin, Fedor V. Philip, Geevarghese Golovach, Petr A. Panolan, Fahad Saurabh, Saket
Author_xml	– sequence: 1 givenname: Fedor V. surname: Fomin fullname: Fomin, Fedor V. organization: University of Bergen – sequence: 2 givenname: Petr A. orcidid: 0000-0002-2619-2990 surname: Golovach fullname: Golovach, Petr A. email: Petr.Golovach@uib.no organization: University of Bergen – sequence: 3 givenname: Fahad surname: Panolan fullname: Panolan, Fahad organization: IIT Hyderabad – sequence: 4 givenname: Geevarghese surname: Philip fullname: Philip, Geevarghese organization: Chennai Mathematical Institute, UMI ReLaX – sequence: 5 givenname: Saket surname: Saurabh fullname: Saurabh, Saket organization: University of Bergen, Institute of Mathematical Sciences
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Snippet	We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems. The input to the Weighted Diverse... We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems. The input to the Weighted Diverse...
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SubjectTerms	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
Title	Diverse collections in matroids and graphs
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