Diverse Collections in Matroids and Graphs

TL;DR

This paper studies the computational complexity of finding diverse solution sets in matroids and graphs, introducing new NP-hard problems and providing fixed-parameter algorithms for them.

Contribution

It defines new diverse solution problems in matroid and graph theory and develops fixed-parameter tractable algorithms for these problems.

Findings

Weighted Diverse Bases and Common Independent Sets are NP-hard.

FPT algorithms are developed for all three problems with parameters (k,d).

The work extends classical matroid and graph problems to diverse solution variants.

Abstract

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 $\omega:E(M)\to\mathbb{N}$, and integers $k\geq 1, d\geq 0$. The task is to decide if there is a collection of $k$ bases $B_{1}, \dotsc, B_{k}$ 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 $\omega:E\to\mathbb{N}$, and integers $k\geq 1, d\geq 0$. The task is to decide if there is a collection of $k$ common independent sets $I_{1}, \dotsc, I_{k}$ 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 $k\geq 1, d\geq 0$. The task is to decide if $G$ contains $k$ perfect matchings $M_{1},\dotsc,M_{k}$ 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.