## 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′ ⊆ 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.

Original language | English |
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Title of host publication | Computing and Combinatorics - 21st International Conference, COCOON 2015, Proceedings |

Editors | Dachuan Xu, Donglei Du, Dingzhu Du |

Publisher | Springer Verlag |

Pages | 548-558 |

Number of pages | 11 |

ISBN (Print) | 9783319213972 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |

Event | 21st International Conference on Computing and Combinatorics Conference, COCOON 2015 - Beijing, China Duration: 4 Aug 2015 → 6 Aug 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9198 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st International Conference on Computing and Combinatorics Conference, COCOON 2015 |
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Country/Territory | China |

City | Beijing |

Period | 4/08/15 → 6/08/15 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science