Showing papers in "Discrete and Computational Geometry in 2012"

An Incidence Theorem in Higher Dimensions

Abstract: We prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in R d . Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.

Approximation Algorithms for Maximum Independent Set of Pseudo-Disks

Abstract: We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local-search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, which leads to a constant-factor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

Abstract: We present a simple and practical (1+e)-approximation algorithm for the Frechet distance between two polygonal curves in ℝ d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.

Strong Homotopy Types, Nerves and Collapses

Abstract: We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial G-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces.

On the Orlicz Minkowski Problem for Polytopes

Abstract: Quite recently, an Orlicz Minkowski problem has been posed and the existence part of this problem for even measures has been presented. In this paper, the existence part of the Orlicz Minkowski problem for polytopes is demonstrated. Furthermore, we obtain a solution of the Orlicz Minkowski problem for general (not necessarily even) measures.

Sphere and Dot Product Representations of Graphs

Abstract: A graph G is a k-sphere graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the dot product of v i and v j is at least 1. By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k>1. In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. Geom. 9:3–24, 1998). In the latter, this answers a question of Fiduccia et al. (Discrete Math. 181:113–138, 1998). Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k>1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (Efficient Graph Representations, 2003).

The Inverse Moment Problem for Convex Polytopes

Abstract: We present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, assuming knowledge of finitely many of its integral moments. In particular, we show that the vertices of an N-vertex convex polytope in ℝ d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure of degree D), in d+1 distinct directions in general position. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii–Pukhlikov, and Barvinok that arise in the discrete geometry of polytopes, combined with what is variously known as Prony’s method, or the Vandermonde factorization of finite rank Hankel matrices.

Extended Formulations for Polygons

Abstract: The extension complexity of a polytope P is the smallest integer k such that P is the projection of a polytope Q with k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193–205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n)×O(n 2) integer grid with extension complexity $\varOmega (\sqrt{n}/\sqrt{\log n})$.

Topology of Random 2-Complexes

Abstract: We study the Linial–Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p≪n −1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π 1(Y) is free and H 2(Y)=0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p≫n −1/2+ϵ , where ϵ>0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n→∞. We also establish several related results; for example, we show that for p

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Abstract: Recently Guth and Katz ( arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in ℝ d , based on the Stone–Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemeredi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.

A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points

Abstract: Boros and Furedi (for d=2) and Barany (for arbitrary d) proved that there exists a positive real number c d such that for every set P of n points in R d in general position, there exists a point of R d contained in at least $c_{d}\binom{n}{d+1}$ d-simplices with vertices at the points of P. Gromov improved the known lower bound on c d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov’s approach and thereby provide a new stronger lower bound on c d for arbitrary d. In particular, we improve the lower bound on c 3 from 0.06332 to more than 0.07480; the best upper bound known on c 3 being 0.09375.

Moving Out the Edges of a Lattice Polygon

Abstract: We review previous work of (mainly) Koelman, Haase and Schicho, and Poonen and Rodriguez-Villegas on the dual operations of (i) taking the interior hull and (ii) moving out the edges of a two-dimensional lattice polygon. We show how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. We then report on an implementation of this algorithm, by means of which we produce the list of all lattice polygons (up to equivalence) whose genus is contained in {1,…,30}. In particular, we obtain the number of inequivalent lattice polygons for each of these genera. As a byproduct, we prove that the minimal possible genus for a lattice 15-gon is 45.

Optimal Topological Simplification of Discrete Functions on Surfaces

Abstract: Given a function f on a surface and a tolerance δ>0, we construct a function f δ subject to ‖f δ −f‖∞≤δ such that f δ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the resulting simplified function f δ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed.

A Non-linear Lower Bound for Planar Epsilon-nets

Abstract: We show that the minimum possible size of an e-net for point objects and line (or rectangle)-ranges in the plane is (slightly) bigger than linear in $\frac{1}{\epsilon}$. This settles a problem raised by Matousek, Seidel and Welzl (Proc. 6th Annu. ACM Sympos. Comput. Geom., pp. 16–22, 1990).

Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Abstract: Let ${\textnormal {R}}$ be a real closed field, $\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]$ finite subsets of polynomials, with the degrees of the polynomials in $\mathcal{P}$ (resp., $\mathcal{Q}$ ) bounded by d (resp., d 0). Let $V \subset {\textnormal {R}}^{k}$ be the real algebraic variety defined by the polynomials in $\mathcal{Q}$ and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family $\mathcal{P}$ on V is bounded by $$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$$ where $s = \operatorname {card}\mathcal{P}$, and $$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$$ In case 2d 0≤d, the above bound can be written simply as $$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$$ (in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d 0≪d) the best known bound of $$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$$ on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d 0. The distinction between the bound d 0 on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in $\mathcal{P}$ that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1[math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1[math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2[math.CO], 2011; Zahl in arXiv:1104.4987v3[math.CO], 2011).

Characteristics of Graph Braid Groups

Abstract: We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.

Some Properties of Gromov–Hausdorff Distances

Abstract: The Gromov–Hausdorff distance between metric spaces appears to be a useful tool for modeling some object matching procedures. Since its conception it has been mainly used by pure mathematicians who are interested in the topology generated by this distance, and quantitative consequences of the definition are not very common. As a result, only few lower bounds for the distance are known, and the stability of many metric invariants is not understood. This paper aims at clarifying some of these points by proving several results dealing with explicit lower bounds for the Gromov–Hausdorff distance which involve different standard metric invariants. We also study a modified version of the Gromov–Hausdorff distance which is motivated by practical applications and both prove a structural theorem for it and study its topological equivalence to the usual notion. This structural theorem provides a decomposition of the modified Gromov–Hausdorff distance as the supremum over a family of pseudo-metrics, each of which involves the comparison of certain discrete analogues of curvature. This modified version relates the standard Gromov–Hausdorff distance to the work of Boutin and Kemper, and Olver.

A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem

Abstract: A short and almost elementary proof of the Boros–Furedi–Barany–Pach–Gromov theorem on the multiplicity of covering by simplices in ℝ d is given.

Matroid Enumeration for Incidence Geometry

Abstract: Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ3 and that of Motzkin on ≤12 lines in ℝ2, extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.

Multitriangulations, Pseudotriangulations and Primitive Sorting Networks

Abstract: We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.

Triangle Contact Representations and Duality

Abstract: A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A primal–dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal–dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.

On ≤ k -Edges, Crossings, and Halving Lines of Geometric Drawings of K n

Abstract: Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$, is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0≤k≤n/2−1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ≤k-edges of P is denoted by E ≤k (P). Let $\overline {\mathrm {cr}}(n)$, h(n), and E ≤k (n) denote the minimum of $\operatorname {cr}(P)$, the maximum of h(P), and the minimum of E ≤k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E ≤k (n) is tight for k<⌈(4n−2)/9⌉ and improve it for all k≥⌈(4n−2)/9⌉. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$. We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n≤27. Exact values were known only for n≤18 and odd n≤21 for the crossing number, and for n≤14 and odd n≤21 for halving lines.

A Hyperplane Inequality for Measures of Convex Bodies in ℝ n , n ≤4

Abstract: Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ n and any origin-symmetric convex body K in ℝ n , $$\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n},$$ where ξ ⊥ is the central hyperplane in ℝ n perpendicular to ξ, and $|B_{2}^{n}|$ is the volume of the unit Euclidean ball in ℝ n . This inequality is sharp, and it generalizes the hyperplane inequality in dimensions up to four to the setting of arbitrary measures in place of volume. In order to prove this inequality, we first establish stability in the affirmative case of the Busemann–Petty problem for arbitrary measures in the following sense: if e>0, K and L are origin-symmetric convex bodies in ℝ n , n≤4, and $$\mu\bigl(K\cap\xi^\bot\bigr) \le\mu\bigl(L\cap\xi^\bot\bigr) +\varepsilon,\quad \forall\xi\in S^{n-1},$$ then $$\mu(K)\le\mu(L) + \frac{n}{n-1}\frac{|B_2^n|^{\frac {n-1}{n}}}{|B_2^{n-1}|} \operatorname{Vol}_n(K)^{1/n}\varepsilon.$$

The Strong Thirteen Spheres Problem

Abstract: The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.

Notes About the Carathéodory Number

Abstract: In this paper we give sufficient conditions for a compactum in ℝ n to have Caratheodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Caratheodory theorem and give a Tverberg-type theorem for families of convex compacta.

Hinged Dissections Exist

Abstract: We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). Our proofs are constructive, giving explicit algorithms in all cases. For two planar polygons whose vertices lie on a rational grid, both the number of pieces and the running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.

Conflict-Free Coloring for Rectangle Ranges Using O ( n .382 ) Colors

Abstract: Given a set of points P⊆ℝ2, a conflict-free coloring of P w.r.t. rectangle ranges is an assignment of colors to points of P, such that each nonempty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P∩T. This notion has been the subject of recent interest and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to base stations (points) such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in ℝ2 can be conflict-free colored with $O(n^{\beta^{*}+o(1)})$ colors in expected polynomial time, where $\beta^{*}=\frac{3-\sqrt{5}}{2} < 0.382$.

Asymmetry of Convex Bodies of Constant Width

Abstract: The symmetry of convex bodies of constant width is discussed in this paper. We proved that for any convex body K⊂ℝn of constant width, $1\leq \mathrm{as}_{\infty}(K)\leq\frac{n+\sqrt{2n(n+1)}}{n+2}$, where as∞(⋅) denotes the Minkowski measure of asymmetry for convex bodies. Moreover, the equality holds on the left-hand side precisely iff K is an Euclidean ball and the upper bounds are attainable, in particular, if n=3, the equality holds on the right-hand side if K is a Meissner body.

A Generalisation of Tverberg’s Theorem

Abstract: We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝ d of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A 1,A 2,…,A k such that for any C⊂S of at most r points, the convex hulls of A 1\C,A 2\C,…,A k \C are intersecting. This was conjectured first by Natalia Garcia-Colin (Ph.D. thesis, University College of London, 2007).

Octants Are Cover-Decomposable

Abstract: We prove that octants are cover-decomposable; i.e., any 12-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into two coverings. As a corollary, we obtain that any 12-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into two coverings. We also show that any 12-fold covering of the whole plane with the translates of a given open triangle can be decomposed into two coverings. However, we exhibit an indecomposable 3-fold covering with translates of a given triangle.