  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.
