Bicriteria Approximation Algorithms for Priority Matroid Median

TL;DR

This paper introduces bicriteria approximation algorithms for the Priority Matroid Median problem, balancing facility selection costs and client radius constraints, with improved results for uniform radius cases.

Contribution

It presents the first bicriteria approximation algorithms for the Priority Matroid Median problem, extending prior work and improving bounds for uniform radius scenarios.

Findings

Bicriteria approximation algorithms with constant factors for the problem.

Improved approximation bounds for uniform radius cases.

Theoretical guarantees on constraint violations and cost bounds.

Abstract

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 ($r_j := L$ $\forall j \in \mathcal{C}$).