Unique covering problems with geometric sets

The Exact Cover problem takes a universe U of n elements, a family F of m subsets of U and a positive integer k, and decides whether there exists a subfamily(set cover) 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 F′ ⊆ F such that at least k of the elements 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 complexitytheoretic assumptions. 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 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 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.