Approximation Algorithms for Fair k-median Problem without Fairness Violation

Abstract

The fair k-median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fairness violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. To our best knowledge, there is no available approximation algorithm for this problem without any fairness violation in doubling metrics. In this paper, we consider the fair k-median problem in doubling metrics and general metrics. We provide the first QPTAS for the fair k-median problem based on the hierarchical decomposition and dynamic programming process in doubling metrics. For applying a dynamic programming process to solve this problem, the distances from portals to facilities cannot be directly enumerated since each client may not be assigned to its closest open facility. To overcome the difficulties caused by the fairness constraints, we construct an auxiliary graph and use minimum weighted perfect matching to get the cost between the portals of each block and the ones in its children. In order to satisfy the fairness constraints, we bound the fairness constraints of each open facility in the leaves of the split-tree based on the relation between the subproblem and the subproblems of its children. To obtain the assignment of the given instance and remove the fairness violation, we construct a new b-value min-cost max-flow model based on the set of open facilities. For the fair k-median problem in general metrics, we present a polynomial-time approximation algorithm with ratio O(log⁡k). Our approximation algorithm for the fair k-median problem in doubling metrics is the first result for the corresponding problem without any fairness violation in doubling metrics.