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# Bicriteria Approximation Algorithms for Priority Matroid Median

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## Cite As

Tanvi Bajpai and Chandra Chekuri. Bicriteria Approximation Algorithms for Priority Matroid Median. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.7

## 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 ℱ and a set of clients 𝒞 that lie in a metric space (ℱ ∪ 𝒞,d), and a matroid ℳ = (ℱ,ℐ) over the facilities. In addition, each client j has a specified radius r_j ≥ 0 and each facility i ∈ ℱ has an opening cost f_i > 0. The goal is to choose a subset S ⊆ ℱ of facilities to minimize ∑_{i ∈ ℱ} f_i + ∑_{j ∈ 𝒞} d(j,S) subject to two constraints: (i) S is an independent set in ℳ (that is S ∈ ℐ) and (ii) for each client j, its distance to an open facility is at most r_j (that is, d(j,S) ≤ r_j). For this problem we describe the first bicriteria (c₁,c₂) approximations for fixed constants c₁,c₂: the radius constraints of the clients are violated by at most a factor of c₁ and the objective cost is at most c₂ times the optimum cost. We also improve the previously known bicriteria approximation for the uniform radius setting (r_j : = L ∀ j ∈ 𝒞).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Facility location and clustering
##### Keywords
• k-median
• fair clustering
• matroid

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