Diverse Collections in Matroids and Graphs
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We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. 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≥ 0. 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. This is a diverse variant of the classical matroid base packing problem. 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≥ 0. 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. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph G and integers k≥ 1, d≥ 0. 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 Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard, and derive fixed-parameter tractable (FPT) algorithms for all three problems with (k,d) as the parameter.
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