### Abstract

The input to the NP-hard Point Line Cover problem (PLC) consists of a set V of n points on the plane and a positive integer k, and the question is whether there exists a set of at most k lines which pass through all points in V. By straightforward reduction rules one can efficiently reduce any input to one with at most k^{2} points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε > 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k^{2-ε}) points. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that-unless the polynomial hierarchy collapses-PLC has no kernel of total size O(k^{2-ε}) bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with n points requires ω(n ^{2}) bits. To get around this hurdle we build on work of Goodman, Pollack and Sturmfels (STOC 1989) and devise an oracle communication protocol of cost O(n log n) for PLC; its main building blocks are a bound of O(N ^{O(n)}) for the order types of n points that are not necessarily in general position and an explicit (albeit slow) algorithm that enumerates a superset of size N^{O(n)} of all possible order types of n points. This protocol, together with the lower bound on the total size (which also holds for such protocols), yields the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is-to the best of our knowledge-the first to show a nontrivial lower bound for structural/secondary parameters.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1596-1606 |

Number of pages | 11 |

ISBN (Print) | 9781611973389 |

State | Published - Jan 1 2014 |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Other

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
---|---|

Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 1596-1606). Association for Computing Machinery.

**Point line cover : The easy kernel is essentially tight.** / Kratsch, Stefan; Philip, Geevarghese; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014.*Association for Computing Machinery, pp. 1596-1606, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, OR, United States, 1/5/14.

}

TY - GEN

T1 - Point line cover

T2 - The easy kernel is essentially tight

AU - Kratsch, Stefan

AU - Philip, Geevarghese

AU - Ray, Saurabh

PY - 2014/1/1

Y1 - 2014/1/1

N2 - The input to the NP-hard Point Line Cover problem (PLC) consists of a set V of n points on the plane and a positive integer k, and the question is whether there exists a set of at most k lines which pass through all points in V. By straightforward reduction rules one can efficiently reduce any input to one with at most k2 points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε > 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k2-ε) points. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that-unless the polynomial hierarchy collapses-PLC has no kernel of total size O(k2-ε) bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with n points requires ω(n 2) bits. To get around this hurdle we build on work of Goodman, Pollack and Sturmfels (STOC 1989) and devise an oracle communication protocol of cost O(n log n) for PLC; its main building blocks are a bound of O(N O(n)) for the order types of n points that are not necessarily in general position and an explicit (albeit slow) algorithm that enumerates a superset of size NO(n) of all possible order types of n points. This protocol, together with the lower bound on the total size (which also holds for such protocols), yields the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is-to the best of our knowledge-the first to show a nontrivial lower bound for structural/secondary parameters.

AB - The input to the NP-hard Point Line Cover problem (PLC) consists of a set V of n points on the plane and a positive integer k, and the question is whether there exists a set of at most k lines which pass through all points in V. By straightforward reduction rules one can efficiently reduce any input to one with at most k2 points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε > 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k2-ε) points. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that-unless the polynomial hierarchy collapses-PLC has no kernel of total size O(k2-ε) bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with n points requires ω(n 2) bits. To get around this hurdle we build on work of Goodman, Pollack and Sturmfels (STOC 1989) and devise an oracle communication protocol of cost O(n log n) for PLC; its main building blocks are a bound of O(N O(n)) for the order types of n points that are not necessarily in general position and an explicit (albeit slow) algorithm that enumerates a superset of size NO(n) of all possible order types of n points. This protocol, together with the lower bound on the total size (which also holds for such protocols), yields the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is-to the best of our knowledge-the first to show a nontrivial lower bound for structural/secondary parameters.

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M3 - Conference contribution

SN - 9781611973389

SP - 1596

EP - 1606

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

ER -