A Lottery Model for Center-Type Problems With Outliers

David G. Harris, Thomas W. Pensyl, Aravind Srinivasan, Khoa Trinh

Abstract: In this article, we give tight approximation algorithms for the k-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client j is allowed to submit a parameter pj ∈ [0,1] and we look for a random solution that covers every client j with probability at least pj. Our techniques include a randomized rounding procedure to round a point inside a matroid intersection polytope to a basis plus at most one extra item such that all marginal probabilities are preserved and such that a certain linear function of the variables does not decrease in the process with probability one.