Speaker: Maryam Negahbani
Affiliation: Dartmouth University

Title: “Revisiting Priority k-Center: Fairness and Outliers

Abstract:
Clustering is a fundamental unsupervised learning and facility location problem extensively studied in the literature. I will talk about a clustering problem called “priority k-center” introduced by Plesnik (in Disc. Appl. Math. 1987). Given a metric space on n points X, with distance function d, an integer k, and radius r_v for each point v in X, the goal is to choose k points S as “centers” to minimize the maximum distance of a point v to S divided by r_v. For uniform r_v’s this is precisely the “k-center” problem where the objective is to minimize the maximum distance of any point to S. In the priority version, points with smaller r_v are prioritized to be closer to S. Recently, a special case of this problem was studied in the context of “individually fair clustering” by Jung et al., FORC 2020. This notion of fairness forces S to open a center in every “densely populated area” by setting r_v to be v’s distance to its closest (n/k)-th neighbor.

In this talk, I show how to approximate priority k-center with outliers: When for a given integer z, you are allowed to throw away z points as outliers and minimize the objective over the rest of the points. We show there is 9-approximation, which is morally a 5, if you have constant many types of radii or if your radii are powers of 2. This is via an LP-aware reduction to min-cost max-flow and is general enough that could handle Matroid constraints on facilities (where instead of asking to pick at most k facilities, you are asked to pick facilities that are independent in a given matroid). Things become quite interesting for priority knapsack-center with outliers: where opening each center costs something and you have a limited budget to spend on your solution S. In this case, we do not know how to solve the corresponding flow problem, so we alter our reduction to reduce to a simpler problem we do know how to solve taking a hit of +5 in the approximation ratio. There are still many open problems in this work, in addition to solving the flow problem in the knapsack case, the best LP integrality gap we know for priority k-center with outliers is 3.