Exact Multi-Covering Problems with Geometric Sets

Abstract

The b-Exact Multicover problem takes a universe U of n elements, a family $\mathcal {F}$ of m subsets of U, a function ${\textsf {dem}}: U \rightarrow \{1,\ldots ,b\}$ and a positive integer k, and decides whether there exists a subfamily(set cover) $\mathcal {F}^{\prime }$ of size at most k such that each element u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$. The b-Exact Coverage problem also takes the same input and decides whether there is a subfamily $\mathcal {F}^{\prime } \subseteq \mathcal {F}$ such that there are at least k elements that satisfy the following property: u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$. Both these problems are known to be NP-complete. In the parameterized setting, when parameterized by k, b-Exact Multicover is W[1]-hard even when b = 1. While b-Exact Coverage is FPT under the same parameter, it is known to not admit a polynomial kernel under standard complexity-theoretic assumptions, even when b = 1. In this paper, we investigate these two problems under the assumption that every set satisfies a given geometric property π. Specifically, we consider the universe to be a set of n points in a real space $\mathbb {R}^{d}$, d being a positive integer. When d = 2 we consider the problem when π requires all sets to be unit squares or lines. When d > 2, we consider the problem where π requires all sets to be hyperplanes in $\mathbb {R}^{d}$. These special versions of the problems are also known to be NP-complete. When parameterized by k, the b-Exact Coverage problem has a polynomial size kernel for all the above geometric versions. The b-Exact Multicover problem turns out to be W[1]-hard for squares even when b = 1, but FPT for lines and hyperplanes. Further, we also consider the b-Exact Max. Multicover problem, which takes the same input and decides whether there is a set cover $\mathcal {F}^{\prime }$ such that every element u ∈ U is covered by at least dem(u) sets and at least k elements satisfy the following property: u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$. To the best of our knowledge, this problem has not been studied before, and we show that it is NP-complete (even for the case of lines). In fact, the problem turns out to be W[1]-hard in the general setting, when parameterized by k. However, when we restrict the sets to lines and hyperplanes, we obtain FPT algorithms.