Point Line Cover: The Easy Kernel is Essentially Tight

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 $P$. A simple polynomial-time reduction reduces any input to one with at most $k^2$ points. We show that this is essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, there is no polynomial-time algorithm that reduces every instance $(P,k)$ of PLC to an equivalent instance with $O(k^{2-\epsilon})$ points, for any $\epsilon>0$. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses 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 PLC---conditionally---has no kernel of total size $O(k^{2-\epsilon})$ 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 $\omega(n^{2})$ bits. To get around this we build on work of Goodman et al. (STOC 1989) and devise an oracle communication protocol of cost $O(n\log n)$ for PLC; its main building block is a bound of $O(n^{O(n)})$ for the order types of $n$ points that are not necessarily in general position, and an explicit algorithm that enumerates all possible order types of n points. This protocol and the lower bound on total size together yield 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

Counting Triangulations and other Crossing-Free Structures Approximately

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of $P$ can be computed in $O^{*}(2^{n})$ time, which is less than the lower bound of $\Omega(2.43^{n})$ on the number of triangulations of any point set. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by $\Lambda$ the output of our algorithm, and by $c^{n}$ the exact number of triangulations of $P$, for some positive constant $c$, we prove that $c^{n}\leq\Lambda\leq c^{n}\cdot 2^{o(n)}$. This is the first algorithm that in sub-exponential time computes a $(1+o(1))$-approximation of the base of the number of triangulations, more precisely, $c\leq\Lambda^{\frac{1}{n}}\leq(1 + o(1))c$. Our algorithm can be adapted to approximately count other crossing-free structures on $P$, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201

An Optimal Generalization of the Colorful Carathéodory Theorem

International audienceThe Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C

A Theorem of Barany Revisited and Extended

International audienceThe colorful Caratheodory theorem states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Caratheodory theorem: given d/2+1 sets of points in $R^d, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a (d/2+1)-dimensional rainbow simplex intersecting C

A simple and less slow method for counting triangulations and for related problems

We present a simple dynamic programming based method for counting straight-edge triangulations of planar point sets. This method can be adapted to solve related problems such as nding the best triangulation of a point set according to certain optimality criteria, or generating a triangulation of a point set uniformly at random. We have implemented our counting method. It appears to be substantially less slow than previous methods: instances with 20 points, which used to take minutes, can now be handled in less than a second, and instances with 30 points, which used to be solvable only by employing several workstations in parallel over a substantial amount of time, an now be solved in about one minute on a single standard workstation.International Max Planck Research Schoo

Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions

In the Set Multicover problem, we are given a set system (X,?), where X is a finite ground set, and ? is a collection of subsets of X. Each element x ? X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection ?\u27 of ? such that each point is covered by at least d(x) sets from ?\u27. In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc. We give a polynomial time (2+?)-approximation algorithm for the set multicover problem (P, ?), where P is a set of points with demands, and ? is a set of non-piercing regions, as well as for the set multicover problem (?, P), where ? is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands

Planar Support for Non-piercing Regions and Applications

Given a hypergraph H=(X,S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S in S, the sub-graph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph H_R(B) = (B, {B_r}_{r in R}), where B_r = {b in B: b cap r != empty set} has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in R cup B. Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions

On a Problem of Danzer

Let C be a bounded convex object in R^d, and P a set of n points lying outside C. Further let c_p, c_q be two integers with 1 <= c_q <= c_p <= n - floor[d/2], such that every c_p + floor[d/2] points of P contains a subset of size c_q + floor[d/2] whose convex-hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex-hull is disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time. In particular, our general theorem implies polynomial bounds for Hadwiger-Debrunner (p, q) numbers for balls in R^d. For example, it follows from our theorem that when p > q >= (1+beta) * d/2 for beta > 0, then any set of balls satisfying the HD(p,q) property can be hit by O(q^2 p^{1+1/(beta)} log p) points. This is the first improvement over a nearly 60-year old exponential bound of roughly O(2^d). Our results also complement the results obtained in a recent work of Keller et al. where, apart from improvements to the bound on HD(p, q) for convex sets in R^d for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane

QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems