  Fairness considerations have motivated new clustering problems and algorithms
in recent years. In this paper we consider the Priority Matroid Median problem
which generalizes the Priority $k$-Median problem that has recently been
studied. The input consists of a set of facilities $\mathcal{F}$ and a set of
clients $\mathcal{C}$ that lie in a metric space $(\mathcal{F} \cup
\mathcal{C},d)$, and a matroid $\mathcal{M}=(\mathcal{F},\mathcal{I})$ over the
facilities. In addition each client $j$ has a specified radius $r_j \ge 0$ and
each facility $i \in \mathcal{F}$ has an opening cost $f_i$. The goal is to
choose a subset $S \subseteq \mathcal{F}$ of facilities to minimize the
$\sum_{i \in \mathcal{F}} f_i + \sum_{j \in \mathcal{C}} d(j,S)$ subject to two
constraints: (i) $S$ is an independent set in $\mathcal{M}$ (that is $S \in
\mathcal{I}$) and (ii) for each client $j$, its distance to an open facility is
at most $r_j$ (that is, $d(j,S) \le r_j$). For this problem we describe the
first bicriteria $(c_1,c_2)$ approximations for fixed constants $c_1,c_2$: the
radius constraints of the clients are violated by at most a factor of $c_1$ and
the objective cost is at most $c_2$ times the optimum cost. We also improve the
previously known bicriteria approximation for the uniform radius setting ($r_j
:= L$ $\forall j \in \mathcal{C}$).
