Fixed-parameter algorithms for Fair Hitting Set problems

Abstract

Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a fair version of Hitting Set. In the classical Hitting Set problem, the input is a universe $U$, a family $F$ of subsets of $U$, and a non-negative integer k. The goal is to determine whether there exists a subset $S\subseteq U$ of size k that hits (i.e., intersects) every set in $F$. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family $B$ of subsets of $U$, where each subset in $B$ can be thought of as the group of elements of the same type. We want to find a set $S\subseteq U$ of size k that (i) hits all sets of $F$, and (ii) does not contain too many elements of each type. We call this problem Fair Hitting Set, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernel for Hitting Set.