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# Unique Covering Problems with Geometric Sets

Ashok, Pradeesha and Kolay, Sudeshna and Misra, Neeldhara and Saurabh, Saket (2015) Unique Covering Problems with Geometric Sets. In: 21st International Computing and Combinatorics Conference (COCOON) , AUG 04-06, 2015, Beijing, PEOPLES R CHINA, pp. 548-558.

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Official URL: http://dx.doi.org/10.1007/978-3-319-21398-9_43

## Abstract

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' subset of 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 W1]-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 R-d, d being a positive integer. When d = 2 we consider the problem when. requires all sets to be unit squares or lines. When d > 2, we consider the problem where. 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 W1]-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 W1]-hard in the abstract setting, when parameterized by k. However, when we restrict ourselves to the lines and hyperplanes versions, we obtain FPT algorithms.

Item Type: Conference Proceedings Publisher Copy right for this article belongs to the SPRINGER-VERLAG BERLIN, HEIDELBERGER PLATZ 3, D-14197 BERLIN, GERMANY Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 24 Nov 2015 06:03 24 Nov 2015 06:03 http://eprints.iisc.ernet.in/id/eprint/52824

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