A better upper bound on the number of triangulations of a planar point set

We show that a point set of cardinality $n$ in the plane cannot be the vertex set of more than $59^n O(n^{-6})$ straight-edge triangulations of its convex hull. This improves the previous upper bound of $276.75^n$.Comment: 6 pages, 1 figur

Upper Bound on the Number of Vertices of Polyhedra with 0,10,1-Constraint Matrices

In this note we show that the maximum number of vertices in any polyhedron $P=\{x\in \mathbb{R}^d : Ax\leq b\}$ with $0,1$-constraint matrix $A$ and a real vector $b$ is at most $d!$.Comment: 3 page

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

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

New lower bounds for the number of straight-edge triangulations of a planar point set

We present new lower bounds on the number of straight-edge triangulations that every set of n points in plane must have. These bounds are better than previous bounds in case of sets with either many or few extreme points

08081 Abstracts Collection -- Data Structures

From February 17th to 22nd 2008, the Dagstuhl Seminar 08081 ``Data Structures\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. It brought together 49 researchers from four continents to discuss recent developments concerning data structures in terms of research but also in terms of new technologies that impact how data can be stored, updated, and retrieved. During the seminar a fair number of participants presented their current research. There was discussion of ongoing work, and in addition an open problem session was held. This paper first describes the seminar topics and goals in general, then gives the minutes of the open problem session, and concludes with abstracts of the presentations given during the seminar. Where appropriate and available, links to extended abstracts or full papers are provided

10091 Abstracts Collection -- Data Structures

From February 28th to March 5th 2010, the Dagstuhl Seminar 10091 "Data Structures" was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. It brought together 45 international researchers to discuss recent developments concerning data structures in terms of research, but also in terms of new technologies that impact how data can be stored, updated, and retrieved. During the seminar a fair number of participants presented their current research and open problems where discussed. This document first briefly describes the seminar topics and then gives the abstracts of the presentations given during the seminar

Inserting one edge into a simple drawing is hard

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G + e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment s, it can be decided in polynomial time whether there exists a pseudocircle Fs extending s for which A ¿ {Fs} is again an arrangement of pseudocircles.Peer ReviewedPostprint (published version

Between umbra and penumbra

International audienceComputing shadow boundaries is a difficult problem in the case of non-point light sources. A point is in the umbra if it does not see any part of any light source; it is in full light if it sees entirely all the light sources; otherwise, it is in the penumbra. While the common boundary of the penumbra and the full light is well understood, less is known about the boundary of the umbra. In this paper we prove various bounds on the complexity of the umbra and the penumbra cast by a segment or polygonal light source on a plane in the presence of polygon or polytope obstacles. In particular, we show that a single segment light source may cast on a plane, in the presence of two triangles, four connected components of umbra and that two fat convex obstacles of total complexity n can engender Omega(n) connected components of umbra. In a scene consisting of a segment light source and k disjoint polytopes of total complexity n, we prove an Omega(nk^2+k^4) lower bound on the maximum number of connected components of the umbra and a O(nk^3) upper bound on its complexity. We also prove that, in the presence of k disjoint polytopes of total complexity n, some of which being light sources, the umbra cast on a plane may have Omega(n^2k^3 + nk^5) connected components and has complexity O(n^3k^3). These are the first bounds on the size of the umbra in terms of both k and n. These results prove that the umbra, which is bounded by arcs of conics, is intrinsically much more intricate than the full light/penumbra boundary which is bounded by line segments and whose worst-case complexity is in Omega(n alpha(k) +km +k^2) and O(n alpha(k) +km alpha(k) +k^2), where m is the complexity of the polygonal light source

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements