Bicriteria Approximation Algorithms for Priority Matroid Median
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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. The goal is to choose a subset S ⊆ℱ of facilities to minimize the ∑_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 (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 ∀ j ∈𝒞).
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