Point Line Cover: The Easy Kernel is Essentially Tight

Abstract

The input to the NP-hard Point Line Cover problem (PLC) consists of a set P 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 P. 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.

Geometric Hitting Set for Segments of Few Orientations

“…(See [20,31].) If the lines have only 2 slopes, then an optimal algorithm is given by the greedy selection of hitting points: Add to the hitting set (initially empty) any point at the intersection of two unhit lines; if no such point exists, and there are still unhit lines, then add to the hitting set a point on an unhit line.…”

“…There is a wealth of related work on geometric set cover and hitting set problems; we do not attempt here to give an exhaustive survey. The point line cover (PLC) problem (see [27,31]) asks for a smallest set of lines to cover a given set of points; it is equivalent, via point-line duality, to the hitting problem for a set of lines. The PLC (and thus the hitting problem for lines) was shown to be NP-hard [34]; in fact, it is APX-hard [11] and Max-SNP Hard [32].…”

New Streaming Algorithms for Parameterized Maximal Matching & Beyond

“…In the parametereized version of the problem, we are given an integer k and the question is whether there exists a set of k lines which can cover all the n points. There is an (known) easy kernel of size O(k 2 ) (see next paragraph), which was shown to be essentially tight by Kratsch et al [3] O(k 2 ) kernel for Point Line Cover: We perform the following procedure iteratively:…”

“…In this paper, we revisit this problem and provide a much simpler algorithm for this problem, which meets the same bounds. We further apply the same technique to the Point Line Cover problem [3] where we are given n points on the plane and a parameter k, and the goal is to see if we can cover all points using k lines. Our belief is that the insights gained from this algorithm could apply to other problems in this area, and that the reduced complexity of the description also makes the algorithm easier to understand and apply.…”

“…In this paper, we revisit this problem and provide a much simpler algorithm for this problem. We are also able to apply the same technique to the Point Line Cover problem [3]. …”