A Lottery Model for Center-type Problems With Outliers

In this paper, 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 $p_j \in [0,1]$ and we look for a random solution that covers every client $j$ with probability at least $p_j$. 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.