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

The Complexity of Guarding Monotone Polygons

Abstract A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)¿k, and we prove that for k=5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time. © 2021 Elsevier B.V

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, $$\frac{40\lfloor (k-1)/2 \rfloor -8}{19} \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11.$$ Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic

Guarding Lines and 2-Link Polygons is APX-hard

We prove that the minimum line covering problem and the minimum guard covering problem restricted to 2-link polygons are APX-hard

Guarding Lines and 2-Link Polygons is APX-hard

We prove that the minimum line covering problem and the minimum guard covering problem restricted to 2-link polygons are APX-hard