Diverse collections in matroids and graphs

• 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 2024; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorial analysis; Combinatorics; Full Length Paper; Integers; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Parameters; Polynomials; Theoretical; Weighting functions; 68Q25; 05B35; 68W40; Diversity of solutions; 05C85; 68Q27; Matroids; Graphs; 05C70; Parameterized complexity
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-023-01959-z

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 .