Showing papers in "Discrete and Computational Geometry in 2018"

Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

Abstract: We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.

Sampling from a Log-Concave Distribution with Projected Langevin Monte Carlo

Abstract: We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O(n 7) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O(n 4) was proved by Lovasz and Vempala.

Berline–Vergne Valuation and Generalized Permutohedra

Abstract: Generalizing a conjecture by De Loera et al., we conjecture that integral generalized permutohedra all have positive Ehrhart coefficients. Berline and Vergne construct a valuation that assigns values to faces of polytopes, which provides a way to write Ehrhart coefficients of a polytope as positive sums of these values. Based on available results, we pose a stronger conjecture: Berline–Vergne’s valuation is always positive on permutohedra, which implies our first conjecture. This article proves that our strong conjecture on Berline–Vergne’s valuation is true for dimension up to 6, and is true if we restrict to faces of codimension up to 3. In addition to investigating the positivity conjectures, we study the Berline–Vergne’s valuation, and show that it is the unique construction for McMullen’s formula used to describe number of lattice points in permutohedra under certain symmetry constraints. We also give an equivalent statement to the strong conjecture in terms of mixed valuations.

Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

Abstract: In this paper we consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ composed of points lying on a $d$-dimensional submanifold $M$, with or without outliers drawn in the ambient space, is observed. Based on the tangential Delaunay complex, we construct an estimator $\hat{M}$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. $\hat{M}$ is built from existing algorithms. In a model without outliers, we show that this estimator is asymptotically minimax optimal for the Hausdorff distance over a class of submanifolds with reach condition. Therefore, even with no a priori information on the tangent spaces of $M$, our estimator based on tangential Delaunay complexes is optimal. This shows that the optimal rate of convergence can be achieved through existing algorithms. A similar result is also derived in a model with outliers. A geometric interpolation result is derived, showing that the tangential Delaunay complex is stable with respect to noise and perturbations of the tangent spaces. In the process, a denoising procedure and a tangent space estimator both based on local principal component analysis (PCA) are studied.

Integral Homology of Random Simplicial Complexes

Abstract: The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as $$n\rightarrow \infty $$ , the first homology group over $$\mathbb {Z}$$ vanishes at the very moment when all the edges are covered by triangular faces.

Tropical Effective Primary and Dual Nullstellensätze

Abstract: Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows studying properties of mathematical objects such as algebraic varieties from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz, and moreover, we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.

Embedding-Preserving Rectangle Visibility Representations of Nonplanar Graphs

Abstract: A (weak) rectangle visibility representation, or simply an RVR, of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary

Abstract: Given a convex body K with smooth boundary \(\partial K\), select a fixed number n of uniformly distributed random points from \(\partial K\). The convex hull \(K_n\) of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of \(K_n\) to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

Abstract: In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid poly-topes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et. al. and present an explicit formula of the Ehrhart polynomial for one of them.

Simplicial Faces of the Set of Correlation Matrices

Abstract: This paper concerns the facial geometry of the set of n×n correlation matrices. The main result states that almost every set of r vertices generates a simplicial face, provided that r ≤ √cn, where c is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than √2n vertices.

Enumeration of lattice 3-polytopes by their number of lattice points

Abstract: We develop a procedure for the complete computational enumeration of lattice 3-polytopes of width larger than one, of which there are finitely many for each given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most 11 lattice points (there are 216,453 of them). In order to achieve this we prove that if P is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enumerate them completely with computer help. Spiked polytopes are infinitely many but admit a quite precise description (and enumeration). Merged polytopes are computed as a union (merging) of two polytopes of width larger than one and strictly smaller number of lattice points.

Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

Abstract: We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.

The Kneser–Poulsen Conjecture for Special Contractions

Abstract: The Kneser–Poulsen conjecture states that if the centers of a family of N unit balls in $${\mathbb E}^d$$ are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $$N\ge (1+\sqrt{2})^d$$ . Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.

On the Circuit Diameter Conjecture

Abstract: From the point of view of optimization, a critical issue is relating the combinatorial diameter of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and Walkup (Acta Math 117:53–78, 1967), the Hirsch conjecture of an upper bound of $$f-d$$ was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron $$U_4$$ with eight facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos (Ann Math 176(1):383–412, 2012). We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the polyhedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equivalences in Klee and Walkup, the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra—in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture. These results offer some hope that the circuit version of the Hirsch conjecture may hold, even for unbounded polyhedra. A challenge in the circuit setting is that different realizations of polyhedra with the same combinatorial structure may have different diameters. We adapt the notion of simplicity to work with circuits in the form of $$\mathcal {C}$$ -simple and wedge-simple polyhedra. We show that it suffices to consider such polyhedra for studying circuit analogues of the Hirsch conjecture.

A Solution of the Erdős–Ulam Problem on Rational Distance Sets Assuming the Bombieri–Lang Conjecture

Abstract: A rational distance set in the plane is a point set which has the property that all pairwise distances between its points are rational. Erdős and Ulam conjectured in 1945 that there is no dense rational distance set in the plane. In this paper we associate an algebraic surface in $${\mathbb {P}}^3$$ , that we call a distance surface, to any finite rational distance set in the plane. Under a mild condition, we prove that a distance surface is always a surface of general type. From this, we deduce that the Bombieri–Lang conjecture in arithmetic algebraic geometry (restricted to the classes of surfaces) implies an answer to the Erdős–Ulam problem. Combined with the results of Solymosi and de Zeeuw, our proofs lead to the following stronger statement: for S a rational distance set with infinitely many points, we have

An Isoperimetric Inequality for Planar Triangulations

Abstract: We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.

Limit Theorems for Random Cubical Homology

Abstract: This paper studies random cubical sets in $$\mathbb {R}^d$$ . Given a cubical set $$X \subset \mathbb {R}^d$$ , a random variable $$\omega _Q\in [0,1]$$ is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with $$\omega _Q\le t$$ for each $$t\in [0,1]$$ . Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

Fast Binary Embeddings with Gaussian Circulant Matrices: Improved Bounds

Abstract: We consider the problem of encoding a finite set of vectors into a small number of bits while approximately retaining information on the angular distances between the vectors. By deriving improved variance bounds related to binary Gaussian circulant embeddings, we largely fix a gap in the proof of the best known fast binary embedding method. Our bounds also show that well-spreadness assumptions on the data vectors, which were needed in earlier work on variance bounds, are unnecessary. In addition, we propose a new binary embedding with a faster running time on sparse data.

Loops in Reeb Graphs of n -Manifolds

Abstract: The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface.

Packing and Covering with Non-Piercing Regions

Abstract: In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). Earlier, PTASs were known only in the setting where the regions were disks. These techniques relied heavily on the circularity of the disks. We develop new techniques to show that a simple local search algorithm yields a PTAS for the problems on non-piercing regions. We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded by some constant. Our result settles a conjecture of Har-Peled from 2014 in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane. Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity. This extends a result of Ene et al. from 2012.

Incidences Between Points and Lines on Two- and Three-Dimensional Varieties

Abstract: Let P be a set of m points and L a set of n lines in $${\mathbb {R}}^4,$$R4, such that the points of P lie on an algebraic three-dimensional variety of degree $$D$$D that does not contain hyperplane or quadric components (a quadric is an algebraic variety of degree two), and no 2-flat contains more than s lines of L. We show that the number of incidences between P and L is $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + nD + m\bigr ) \end{aligned}$$I(P,L)=O(m1/2n1/2D+m2/3n1/3s1/3+nD+m)for some absolute constant of proportionality. This significantly improves the bound of the authors (Sharir, Solomon, Incidences between points and lines in $${\mathbb {R}}^4.$$R4. Discrete Comput Geom 57(3), 702---756, 2017), for arbitrary sets of points and lines in $${\mathbb {R}}^4,$$R4, when $$D$$D is not too large. Moreover, when $$D$$D and s are constant, we get a linear bound. The same bound holds when the three-dimensional surface is embedded in any higher-dimensional space. The bound extends (with a slight deterioration, when $$D$$D is large) to the complex field too. For a complex three-dimensional variety, of degree $$D,$$D, embedded in $${\mathbb {C}}^4$$C4 (or in any higher-dimensional $${\mathbb {C}}^d$$Cd), under the same assumptions as above, we have $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + D^6 + nD + m \bigr ). \end{aligned}$$I(P,L)=O(m1/2n1/2D+m2/3n1/3s1/3+D6+nD+m).For the proof of these bounds, we revisit certain parts of [36], combined with the following new incidence bound, for which we present a direct and fairly simple proof. Going back to the real case, let P be a set of m points and L a set of n lines in $${\mathbb {R}}^d,$$Rd, for $$d\ge 3,$$dź3, which lie in a common two-dimensional algebraic surface of degree $$D$$D that does not contain any 2-flat, so that no 2-flat contains more than s lines of L (here we require that the lines of L also be contained in the surface). Then the number of incidences between P and L is $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + m + n\bigr ). \end{aligned}$$I(P,L)=O(m1/2n1/2D1/2+m2/3D2/3s1/3+m+n).When $$d=3,$$d=3, this improves the bound of Guth and Katz (On the Erdźs distinct distances problem in the plane. Ann Math 181(1), 155---190, 2015) for this special case, when $$D\ll n^{1/2}.$$Dźn1/2. Moreover, the bound does not involve the term O(nD). This term arises in most standard approaches, and its removal is a significant aspect of our result. Again, the bound is linear when $$D= O(1).$$D=O(1). This bound too extends (with a slight deterioration, when $$D$$D is large) to the complex field. For a complex two-dimensional variety, of degree $$D,$$D, when the ambient space is $${\mathbb {C}}^3$$C3 (or any higher-dimensional $${\mathbb {C}}^d$$Cd), under the same assumptions as above, we have $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + D^3 + m + n\bigr ). \end{aligned}$$I(P,L)=O(m1/2n1/2D1/2+m2/3D2/3s1/3+D3+m+n).These new incidence bounds are among the very few bounds, known so far, that hold over the complex field. The bound for two-dimensional (resp., three-dimensional) varieties coincides with the bound in the real case when $$D= O(m^{1/3})$$D=O(m1/3) (resp., $$D= O(m^{1/6})$$D=O(m1/6)).

Extension Complexity and Realization Spaces of Hypersimplices

Abstract: The (n, k)-hypersimplex is the convex hull of all 0 / 1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of certain classes of combinatorial hypersimplices. To that end, we investigate the projective realization spaces of hypersimplices and their (refined) rectangle covering numbers. Our proofs combine ideas from geometry and combinatorics and are partly computer assisted.

Plane Bichromatic Trees of Low Degree

Abstract: Let R and B be two disjoint sets of points in the plane such that $$|B|\le |R|$$ , and no three points of $$R\cup B$$ are collinear. We show that the geometric complete bipartite graph K(R, B) contains a non-crossing spanning tree whose maximum degree is at most $$\max \,\{3, \lceil (|R|-1)/|B|\rceil + 1\}$$ ; this is the best possible upper bound on the maximum degree. This proves two conjectures made by Kaneko, 1998, and solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.

On Sets Defining Few Ordinary Circles

Abstract: An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least $$n^2/4 - O(n)$$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most $$n^3/24 - O(n^2)$$ circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most $$Kn^2$$ ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Any Finite Group is the Group of Some Binary, Convex Polytope

Abstract: For any given finite group, Schulte and Williams (Discrete Comput Geom 54(2):444–458, 2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (J Combin Theory Ser B 31(3):297–312, 1981); the diameter of its skeleton is at most 2; any combinatorial automorphism of the polytope is induced by some isometry of the space; any automorphism of the skeleton is a combinatorial automorphism.

Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions

Abstract: Given n pairwise disjoint non-vertical lines in 3-space, their vertical depth (i.e., above/below) relation may contain cycles. We show that the lines can be cut into $$O(n^{3/2}{{\mathrm{polylog}}}\, n)$$ pieces, such that the depth relation among these pieces is a proper partial order. This bound is nearly tight in the worst case. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. Our technique can be extended to eliminating all cycles in the depth relation among segments and among constant-degree algebraic arcs. Our results almost completely settle a 35-year-old open problem in computational geometry motivated by hidden-surface removal in computer graphics. We also discuss several algorithms for constructing a small set of cuts so as to eliminate all depth-relation cycles among the lines.

Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications

Abstract: We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known. Finally, we extend a result of Barone and Basu on bounding the number of connected components of real varieties defined by two polynomials of differing degrees to the sum of all Betti numbers, thus making progress on an open problem posed in that paper.

On the Number of Maximum Empty Boxes Amidst n Points

Abstract: We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S.

The rigidity of infinite graphs

Abstract: A rigidity theory is developed for countably infinite simple graphs in $${\mathbb {R}}^d$$ . Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $${\mathbb {R}}^2$$ and Tay’s multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $${\mathbb {R}}^d$$ . Analogous results are obtained for the classical non-Euclidean $$\ell ^q$$ norms.

On the Reverse Loomis–Whitney Inequality

Abstract: The present paper deals with the problem of computing (or at least estimating) the $$\mathrm {LW}$$ -number $$\lambda (n)$$ , i.e., the supremum of all $$\gamma $$ such that for each convex body K in $${\mathbb {R}}^n$$ there exists an orthonormal basis $$\{u_1,\ldots ,u_n\}$$ such that $$\begin{aligned} {\text {vol}}_n(K)^{n-1} \ge \gamma \prod _{i=1}^n {\text {vol}}_{n-1} (K|u_i^{\perp }) , \end{aligned}$$ where $$K|u_i^{\perp }$$ denotes the orthogonal projection of K onto the hyperplane $$u_i^{\perp }$$ perpendicular to $$u_i$$ . Any such inequality can be regarded as a reverse to the well-known classical Loomis–Whitney inequality. We present various results on such reverse Loomis–Whitney inequalities. In particular, we prove some structural results, give bounds on $$\lambda (n)$$ and deal with the problem of actually computing the $$\mathrm {LW}$$ -constant of a rational polytope.