Point line cover: The easy kernel is essentially tight

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(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.