Unique Covering Problems with Geometric Sets

Abstract

The Exact Cover problem takes a universe U of n elements, a family $\mathcal F$ of m subsets of U and a positive integer k, and decides whether there exists a subfamily(set cover) $\mathcal F '$ of size at most k such that each element is covered by exactly one set. The Unique Cover problem also takes the same input and decides whether there is a subfamily $\mathcal F ' \subseteq \mathcal F$ such that at least k of the elements $\mathcal F '$ covers are covered uniquely(by exactly one set). Both these problems are known to be NP-complete. In the parameterized setting, when parameterized by k, Exact Cover is W[1]-hard. While Unique Cover is FPT under the same parameter, it is known to not admit a polynomial kernel under standard complexity-theoretic assumptions.

In this paper, we investigate these two problems under the assumption that every set satisfies a given geometric property $\Pi$. 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 $\Pi$ requires all sets to be unit squares or lines. When $d >2$, we consider the problem where $\Pi$ requires all sets to be hyperplanes in ${\mathbb R}^d$. These special versions of the problems are also known to be NP-complete. When parameterizing by k, the Unique Cover problem has a polynomial size kernel for all the above geometric versions. The Exact Cover problem turns out to be W[1]-hard for squares, but FPT for lines and hyperplanes. Further, we also consider the Unique Set Cover problem, which takes the same input and decides whether there is a set cover which covers at least k elements uniquely. To the best of our knowledge, this is a new problem, 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 abstract setting, when parameterized by k. However, when we restrict ourselves to the lines and hyperplanes versions, we obtain FPT algorithms.

N. Misra—Supported by the INSPIRE Faculty Scheme, DST India (project DSTO-1209).