# 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.

## Keywords

Polynomial Kernel Reduction Rule Input Instance Unique Cover Exact Cover## Preview

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