Point line cover: The easy kernel is essentially tight

Stefan Kratsch, Geevarghese Philip, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 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.

Original languageEnglish (US)
Title of host publicationProceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
PublisherAssociation for Computing Machinery
Pages1596-1606
Number of pages11
ISBN (Print)9781611973389
DOIs
StatePublished - 2014
Event25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States
Duration: Jan 5 2014Jan 7 2014

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
CountryUnited States
CityPortland, OR
Period1/5/141/7/14

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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