Parameterized Complexity of d-Hitting Set with Quotas

Abstract

In this paper we study a variant of the classic d -Hitting Set problem with lower and upper capacity constraints, say A and B, respectively. The input to the problem consists of a universe U, a set family, $\mathscr {S}$, of sets over U, where each set in the family is of size at most d, a non-negative integer k; and additionally two functions $\alpha :\mathscr {S} \rightarrow \{1,\ldots ,A\}$ and $\beta :\mathscr {S} \rightarrow \{1,\ldots ,B\}$. The goal is to decide if there exists a hitting set of size at most k such that for every set S in the family $\mathscr {S}$, the solution contains at least $\alpha (S)$ elements and at most $\beta (S)$ elements from S. We call this the $(A, B)$-Multi d-Hitting Set problem. We study the problem in the realm of parameterized complexity. We show that $(A, B)$-Multi d-Hitting Set can be solved in $\mathcal {O}^{\star }(d^{k})$ time. For the special case when $d=3$ and $d=4$, we have an improved bound of $\mathcal {O}^\star (2.2738^k)$ and $\mathcal {O}^\star (3.562^{k})$, respectively. The former matches the running time of the classical 3-Hitting Set problem. Furthermore, we show that if we do not have an upper bound constraint and the lower bound constraint is same for all the sets in the family, say $A>1$, then the problem can be solved even faster than d-Hitting Set.

We next investigate some graph-theoretic problems which can be thought of as an implicit d-Hitting Set problem. In particular, we study $(A, B)$-Multi Vertex Cover and $(A, B)$-Multi Feedback Vertex Set in Tournaments. In $(A, B)$-Multi Vertex Cover, we are given a graph G and a non-negative integer k, the goal is to find a subset $S\subseteq V(G)$ of size at most k such that for every edge in G, S contains at least A and at most B of its endpoints. Analogously, we can define $(A, B)$-Multi Feedback Vertex Set in Tournaments. We show that unlike Vertex Cover, which is same as $(1, 2)$-Multi Vertex Cover, $(1, 1)$-Multi Vertex Cover is polynomial-time solvable. Furthermore, unlike Feedback Vertex Set in Tournaments, $(A, B)$-Multi Feedback Vertex Set in Tournaments can be solved in polynomial time.

Sushmita Gupta was supported by SERB-Starting Research Grant (SRG/2019/001870).