Showing papers in "Discrete and Computational Geometry in 2017"

Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

Abstract: The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss–Bonnet relations, and the principal kinematic formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.

Recognition and Complexity of Point Visibility Graphs

Abstract: A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: Given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points on a grid. We also prove that the problem of recognizing visibility graphs of points on a grid is decidable if and only if the existential theory of the rationals is decidable.

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Abstract: Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Frechet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $$O(n^2 \log n)$$O(n2logn) algorithm by Alt and Godau for computing the Frechet distance remains the state of the art (here, n denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Frechet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Frechet distance between two polygonal curves in time $$O(n^2 \sqrt{\log n}(\log \log n)^{3/2})$$O(n2logn(loglogn)3/2) on a pointer machine and in time $$O(n^2(\log \log n)^2)$$O(n2(loglogn)2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth $$O(n^{2-\varepsilon })$$O(n2-ź), for some $$\varepsilon > 0$$ź>0. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Frechet distance on a word RAM.

Computational Aspects of the Gromov---Hausdorff Distance and its Application in Non-rigid Shape Matching

Abstract: The Gromov---Hausdorff distance of two compact metric spaces is a measure for how far the spaces are from being isometric and has been extensively studied in the field of metric geometry. In recent years, a lot of attention has been devoted to computational aspects of the Gromov---Hausdorff distance. One of the most prominent applications is the problem of shape matching, where the goal is to decide whether two shapes given as polygonal meshes are equivalent under certain invariances. Therefore, many methods have been proposed which heuristically estimate the Gromov---Hausdorff distance of metric spaces induced by the shapes. However, the computational complexity of computing the Gromov---Hausdorff distance has not yet been thoroughly investigated. We show that--under standard complexity theoretic assumptions--determining the Gromov---Hausdorff distance of two finite metric spaces cannot be approximated within any reasonable bound in polynomial time. Furthermore, we discover attributes of metric spaces which have a major impact on the complexity of an instance. This enables us to develop an approximation algorithm which is fixed parameter tractable with respect to corresponding parameters.

Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces

Abstract: The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David’s and Christ’s constructions of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure.

Computing the Flip Distance Between Triangulations

Abstract: Let $$\mathcal{{T}}$$ be a triangulation of a set $$\mathcal{{P}}$$ of n points in the plane, and let e be an edge shared by two triangles in $$\mathcal{{T}}$$ such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of $$\mathcal{{P}}$$ from $$\mathcal{{T}}$$ . The flip distance between two triangulations of $$\mathcal{{P}}$$ is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of $$\mathcal{{P}}$$ is at most k, for some given $$k \in \mathbb {N}$$ . It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time $$\mathcal {O}(n + k \cdot c^{k})$$ , for a constant $$c \le 2 \cdot 14^{11}$$ , which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.

The Threshold for Integer Homology in Random d-Complexes

Abstract: Let $$Y \sim Y_d(n,p)$$Y~Yd(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology $$H_{d-1}(Y; \mathbb {Z})$$Hd-1(YźZ) is less than $$40d (d+1) \log n / n$$40d(d+1)logn/n. This bound is tight, up to a constant factor which depends on d.

New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

Abstract: In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $$l^p_3$$ -norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong’s recent upper bound for the maximal density of translative packings of regular tetrahedra from $$0.3840\ldots $$ to $$0.3745\ldots $$ , getting closer to the best known lower bound of $$0.3673\ldots $$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.

Recognizing Weakly Simple Polygons

Abstract: We present an $$O(n\log n)$$O(nlogn)-time algorithm that determines whether a given n-gon in the plane is weakly simple. This improves upon an $$O(n^2\log n)$$O(n2logn)-time algorithm by Chang et al. (Proceedings of the 26th ACM-SIAM symposium on discrete algorithm, SIAM, 2015). Weakly simple polygons are required as input for several geometric algorithms. As such, recognizing simple or weakly simple polygons is a fundamental problem.

On the Combinatorial Complexity of Approximating Polytopes

Abstract: Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter $$\mathrm {diam}(K)$$ is given in Euclidean d-dimensional space, where d is a constant. Given an error parameter $$\varepsilon > 0$$ , the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most $$\varepsilon \cdot \mathrm {diam}(K)$$ . By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $$O(1/\varepsilon ^{(d-1)/2})$$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $$\widetilde{O}(1/\varepsilon ^{(d-1)/2})$$ , where $$\widetilde{O}$$ conceals a polylogarithmic factor in $$1/\varepsilon $$ . This is a significant improvement upon the best known bound, which is roughly $$O(1/\varepsilon ^{d-2})$$ . Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Barany and Larman’s economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

Quantitative Combinatorial Geometry for Continuous Parameters

Abstract: We prove variations of Caratheodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovasz's colorful Helly's theorem, Barany's colorful Caratheodory's theorem, and the colorful Tverberg's theorem.

SL(n) Invariant Valuations on Polytopes

Abstract: A classification of $${\text {SL}}(n)$$SL(n) invariant valuations on the space of convex polytopes in $$\mathbb {R}^n$$Rn without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $$\mathbb {R}^n$$Rn that contain the origin.

On Homotopy Types of Euclidean Rips Complexes

Abstract: The Rips complex at scale r of a set of points X in a metric space is the abstract simplicial complex whose faces are determined by finite subsets of X of diameter less than r. We prove that for X in the Euclidean 3-space $$\mathbb {R}^3$$ the natural projection map from the Rips complex of X to its shadow in $$\mathbb {R}^3$$ induces a surjection on fundamental groups. This partially answers a question of Chambers, de Silva, Erickson and Ghrist who studied this projection for subsets of $$\mathbb {R}^2$$ . We further show that Rips complexes of finite subsets of $$\mathbb {R}^n$$ are universal, in that they model all homotopy types of simplicial complexes PL-embeddable in $$\mathbb {R}^n$$ . As an application we get that any finitely presented group appears as the fundamental group of a Rips complex of a finite subset of $$\mathbb {R}^4$$ . We furthermore show that if the Rips complex of a finite point set in $$\mathbb {R}^2$$ is a normal pseudomanifold of dimension at least two then it must be the boundary of a crosspolytope.

Quantitative Helly-Type Theorem for the Diameter of Convex Sets

Abstract: We provide a new quantitative version of Helly's theorem: there exists an absolute constant $$\alpha >1$$ź>1 with the following property. If $$\{P_i: i\in I\}$${Pi:iźI} is a finite family of convex bodies in $${\mathbb {R}}^n$$Rn with $${\mathrm{int}} (\bigcap _{i\in I}P_i ) e \emptyset $$int(źiźIPi)źź, then there exist $$z\in {\mathbb {R}}^n$$zźRn, $$s\leqslant \alpha n$$sźźn and $$i_1,\ldots i_s\in I$$i1,źisźI such that $$\begin{aligned} z+P_{i_1}\cap \cdots \cap P_{i_s}\subseteq cn^{3/2}\Big (z+\bigcap _{i\in I}P_i\Big ), \end{aligned}$$z+Pi1źźźPis⊆cn3/2(z+źiźIPi),where $$c>0$$c>0 is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Barany, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound $$O(n^{3/2})$$O(n3/2) can be improved to $$O(\sqrt{n})$$O(n).

New Error Measures and Methods for Realizing Protein Graphs from Distance Data

Abstract: The interval distance geometry problem consists in finding a realization in $$\mathbb {R}^K$$RK of a simple undirected graph $$G=(V,E)$$G=(V,E) with non-negative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver $$+$$+ formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates.

Untangling Planar Curves

Abstract: Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires $$\Theta (n^{3/2})$$ homotopy moves in the worst case. Our algorithm improves the best previous upper bound $$O(n^2)$$ , which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that $$\Omega (n^{3/2})$$ facial electrical transformations are required to reduce any plane graph with treewidth $$\Omega (\sqrt{n})$$ to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires $$\Theta (n^{3/2} + nk + k^2)$$ homotopy moves in the worst case. Finally, we prove that transforming one non-contractible closed curve to another on any orientable surface requires $$\Omega (n^2)$$ homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets

Abstract: This paper presents a new variation of Tverberg's theorem. Given a discrete set S of $$\mathbb {R}^{d}$$Rd, we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly's and Caratheodory's theorems.

Extremal examples of collapsible complexes and random discrete Morse theory

Abstract: We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $$d \ge 2$$ , there are collapsible (and shellable) simplicial d-complexes with only one free face. Also, there are non-evasive d-complexes with only two free faces (both results are optimal in all dimensions). (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible 3-dimensional simplicial complex with face vector $$f=(106,596,1064,573)$$ that admits two distinct optimal discrete Morse vectors, (1, 1, 1, 0) and (1, 0, 1, 1). Indeed, we show that in every dimension $$d\ge 3$$ there are contractible, non-collapsible simplicial d-complexes that have $$(1,0,\dots ,0,1,1,0)$$ and $$(1,0,\dots ,0,0,1,1)$$ as distinct optimal discrete Morse vectors. (3) We give a first explicit example of a (non-PL) 5-manifold, with face vector $$f=(5013,72300,290944,$$ 495912, 383136, 110880), that is collapsible but not homeomorphic to a ball. Furthermore, we discuss possible improvements and drawbacks of random approaches to collapsibility and discrete Morse theory. We will introduce randomized versions random-lex-first and random-lex-last of the lex-first and lex-last discrete Morse strategies of Benedetti and Lutz (Exp Math 23(1):66–94, 2014), respectively—and we will see that in many instances the random-lex-last strategy works significantly better than Benedetti–Lutz’s (uniform) random strategy. On the theoretical side, we prove that after repeated barycentric subdivisions, the discrete Morse vectors found by randomized algorithms have, on average, an exponential (in the number of barycentric subdivisions) number of critical cells asymptotically almost surely.

Algorithmic Solvability of the Lifting-Extension Problem

Abstract: Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and $$\dim X\le 2d$$dimX≤2d, for some $$d\ge 1$$dź1, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps $$|X|\rightarrow |Y|$$|X|ź|Y|; the existence of such a map can be decided even for $$\dim X\le 2d+1$$dimX≤2d+1. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into $$\mathbb R^n$$Rn under the condition $$k\le \frac{2}{3} n-1$$k≤23n-1. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.

Incidences Between Points and Lines in $${\mathbb {R}}^4$$R4

Abstract: We show that the number of incidences between m distinct points and n distinct lines in $${\mathbb {R}}^4$$R4 is $$O(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n)$$O(2clogm(m2/5n4/5+m)+m1/2n1/2q1/4+m2/3n1/3s1/3+n), for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor $$2^{c\sqrt{\log m}}$$2clogm when $$m \le n^{6/7}$$m≤n6/7 or $$m \ge n^{5/3}$$mźn5/3. Except for the factor $$2^{c\sqrt{\log m}}$$2clogm, the bound is tight in the worst case.

Limits of Local Search: Quality and Efficiency

Abstract: Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set $$\mathcal {D}$$D of geometric objects, compute the minimum-sized subset of P that hits all objects in $$\mathcal {D}$$D. For the case where $$\mathcal {D}$$D is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum (SIAM J Comput 11:555---556, 1982), was finally achieved in Mustafa and Ray (Discret Comput Geom 44:883---895, 2010). Surprisingly, the algorithm to achieve the PTAS is simple: local-search. In particular, the algorithm starts with any hitting set, and iteratively tries to decrease its size by trying to replace some k points by $$k-1$$k-1 points; call such an algorithm a $$(k, k-1)$$(k,k-1)-local search algorithm. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. In particular, the current best work shows that if $$k \ge 30$$kź30, then local-search is able to give a constant factor (as a function of k) approximation ratio (Fraser in Algorithms for Geometric Covering and Piercing Problems, 2012). Unfortunately this then implies that the running time for local-search to provably work at all is $$\Omega (n^{30})$$Ω(n30) using the current framework. As currently local search is the only known method that gives approximation factors that could be useful in practice, it becomes important to explore the limits--in both efficiency and quality--of local search. Simple examples show that (1, 0) and (2, 1) local search cannot give constant factor approximations. In this paper, we show that, surprisingly, just (3, 2) local search is able to give a constant-factor approximation; in fact we are able to get the precise quality limit of (3, 2)-local search: factor 8 approximation. This simplest working instance of local search already gives an approximation factor that is better than all known other methods! In fact, our improvement applies to all algorithms that use local-search for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others. Finding efficient (3, 2)-local search algorithms then becomes the key bottleneck in efficient and good-quality algorithms. In this paper we present such improved algorithms.

A Note on the Tolerant Tverberg Theorem

Abstract: The tolerant Tverberg theorem generalizes Tverberg’s theorem by introducing a new parameter t called tolerance. It states that there is a minimal number N so that any set of at least N points in $$\mathbb R^d$$ can be partitioned into r disjoint sets such that they remain intersecting even after removing any t points from X. In this paper we give an asymptotically tight bound for the tolerant Tverberg Theorem when the dimension and the size of the partition are fixed. To achieve this, we study certain partitions of order-type homogeneous sets and use a generalization of the Erdős–Szekeres theorem. As far as we know, this is the first time that a Ramsey-type theorem has been used to prove a Tverberg-type result.

Universal Rigidity of Complete Bipartite Graphs

Abstract: We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $$(K(n,m),\mathbf{p},\mathbf{q})$$(K(n,m),p,q). This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position. Even without any of these assumptions, in complete generality, we extend these ideas to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.

Upper and Lower Bounds for Online Routing on Delaunay Triangulations

Abstract: Consider a weighted graph G whose vertices are points in the plane and edges are line segments between pairs of points whose weight is the Euclidean distance between its endpoints. A routing algorithm on G sends a message from any vertex s to any vertex t in G. The algorithm has a competitive ratio of c if the length of the path taken by the message is at most c times the length of the shortest path from s to t in G. It has a routing ratio of c if the length of the path is at most c times the Euclidean distance from s to t. The algorithm is online if it makes forwarding decisions based on (1) the k-neighborhood in G of the message’s current position (for constant k > 0) and (2) limited information stored in the message header.

Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples

Abstract: Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows approximation of these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

The Realizability of Curves in a Tropical Plane

Abstract: Let E be a plane in an algebraic torus $$ (K^*)^n $$(Kź)n over an algebraically closed field K. Given a balanced 1-dimensional fan C in the tropicalization of E, i. e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic curve in E. In the first important case when E is a general plane in a 3-dimensional torus we also use our algorithm to prove some general criteria for C that imply its realizability resp. non-realizability. They include and generalize the main known obstructions by Brugalle-Shaw and Bogart-Katz coming from tropical intersection theory.

Coloring Points with Respect to Squares

Abstract: We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane $${\mathcal {S}} \subset {\mathbb {R}}^2$$ can be 2-colored such that every axis-parallel square that contains at least m points from $${\mathcal {S}}$$ contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram.

$$\varepsilon $$ź-Mnets: Hitting Geometric Set Systems with Subsets

Abstract: The existence of Macbeath regions is a classical theorem in convex geometry (Macbeath in Ann Math 56:269---293, 1952), with recent applications in discrete and computational geometry. In this paper, we initiate the study of Macbeath regions in a combinatorial setting and establish near-optimal bounds for several basic geometric set systems.

Some Conditionally Hard Problems on Links and 3-Manifolds

Abstract: We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with Thurston norm at most a given integer; this is shown to be NP-complete. The second problem is the homeomorphism problem for closed 3-manifolds; this is shown to be at least as hard as the graph isomorphism problem. The third problem is determining whether a given link in the 3-sphere is a sublink of another given link; this is shown to be NP-hard.

On the Bounds of Conway’s Thrackles

Abstract: A thrackle on a surface X is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that $$e\le n$$ for every thrackle on a sphere. Until now, the best known bound is $$e\le 1.428n$$ . By using discharging rules we show that $$e\le 1.4n-1.4$$ .