Showing papers in "Discrete and Computational Geometry in 2020"

Decomposition of Exact pfd Persistence Bimodules

Abstract: We characterize the class of persistence modules indexed over $$\mathbb {R}^2$$ that are decomposable into summands whose supports have the shape of a block—i.e. a horizontal band, a vertical band, an upper-right quadrant, or a lower-left quadrant. Assuming the modules are pointwise finite dimensional (pfd), we show that they are decomposable into block summands if and only if they satisfy a certain local property called exactness. Our proof follows the same scheme as the proof of decomposition for pfd persistence modules indexed over $$\mathbb {R}$$, yet it departs from it at key stages due to the product order on $$\mathbb {R}^2$$ not being a total order, which leaves some important gaps open. These gaps are filled in using more direct arguments. Our work is motivated primarily by the stability theory for zigzags and interlevel-sets persistence modules, in which block-decomposable bimodules play a key part. Our results allow us to drop some of the conditions under which that theory holds, in particular the Morse-type conditions.

Near-Optimal Coresets of Kernel Density Estimates

Abstract: We construct near-optimal coresets for kernel density estimates for points in $${\mathbb {R}}^d$$ when the kernel is positive definite. Specifically we provide a polynomial time construction for a coreset of size $$O(\sqrt{d}/\varepsilon \cdot \sqrt{\log 1/\varepsilon } )$$ , and we show a near-matching lower bound of size $$\Omega (\min \{\sqrt{d}/\varepsilon , 1/\varepsilon ^2\})$$ . When $$d\ge 1/\varepsilon ^2$$ , it is known that the size of coreset can be $$O(1/\varepsilon ^2)$$ . The upper bound is a polynomial-in- $$(1/\varepsilon )$$ improvement when $$d \in [3,1/\varepsilon ^2)$$ and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

Ideal Hyperbolic Polyhedra and Discrete Uniformization

Abstract: We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmuller spaces $$\widetilde{\mathscr {T}}_{g,n}$$ of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over $$\mathscr {T}_{g,n}$$ , and invariant under the action of the mapping class group.

The chromatic number of the plane is at least 5 - a new proof

Abstract: We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points one unit apart which are identically colored.

Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Abstract: Given a finite range space $$\Sigma =(\mathsf {X},\mathcal {R})$$, with $$N= |\mathsf {X}| + |\mathcal {R}|$$, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of $$\Sigma $$. The first algorithm is a simpler variant of the Bronnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to $$O(N\mathrm {polylog}(N))$$ expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if $$\mathsf {X}$$ is a set of points and $$\mathcal {R}$$ a set of disks in $$\mathbb {R}^2$$.

Optimal Line Packings from Nonabelian Groups

Abstract: We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.

Tree Drawings Revisited

Abstract: We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that (1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area $$n2^{O(\sqrt{\log \log n\log \log \log n})}$$ , improving the longstanding $$O(n\log n)$$ bound; (2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area $$n\sqrt{\log n}(\log \log n)^{O(1)}$$ , improving the longstanding $$O(n\log n)$$ bound; (3) every binary tree of size n has a straight-line orthogonal drawing with area $$n2^{O(\log ^*n)}$$ , improving the previous $$O(n\log \log n)$$ bound; (4) every binary tree of size n has a straight-line order-preserving drawing with area $$n2^{O(\log ^*n)}$$ , improving the previous $$O(n\log \log n)$$ bound; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area $$n2^{O(\sqrt{\log n})}$$ , improving the previous $$O(n^{3/2})$$ bound.

Discretizations of Surfaces with Constant Ratio of Principal Curvatures

Abstract: Motivated by applications in architecture, we study surfaces with a constant ratio of principal curvatures. These surfaces are a natural generalization of minimal surfaces, and can be constructed by applying a Christoffel-type transformation to appropriate spherical curvature line parametrizations, both in the smooth setting and in a discretization with principal nets. We link this Christoffel-type transformation to the discrete curvature theory for parallel meshes and characterize nets that admit these transformations. In the case of negative curvature, we also present a discretization of asymptotic nets. This case is suitable for design and computation, and forms the basis for a special type of architectural support structures, which can be built by bending flat rectangular strips of inextensible material, such as sheet metal.

Reconstructing Plane Quartics from Their Invariants

Abstract: We present an explicit method that, given a generic tuple of Dixmier–Ohno invariants, reconstructs a corresponding plane quartic curve.

Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Abstract: Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for $$m \le n/{\text {polylog}}n$$. Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for $$m \le n/{\text {polylog}}n$$. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications

Abstract: We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $$L_p$$ -norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $$O(n \log ^3 n)$$ storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required $$O(n^{\varepsilon })$$ time for an update and $$O(\log n)$$ time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $$O(n^{\varepsilon })$$ factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in $${{\mathbb {R}}}^3$$ . Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

On Tope Graphs of Complexes of Oriented Matroids

Abstract: We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we obtain purely graph theoretic polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case. In particular, our results answer a relatively long-standing open question in oriented matroids and can be seen as identifying the theory of (complexes of) oriented matroids as a part of metric graph theory. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.

Ordered and convex geometric trees with linear extremal function

Abstract: The extremal functions $$\mathrm{{ex}}_{\rightarrow }(n,F)$$ and $$\mathrm{{ex}}_{\circlearrowright }(n,F)$$ for ordered and convex geometric acyclic graphs F have been extensively investigated by a number of researchers. Basic questions are to determine when $$\mathrm{{ex}}_{\rightarrow }(n,F)$$ and $$\mathrm{{ex}}_{\circlearrowright }(n,F)$$ are linear in n, the latter posed by Brass–Karolyi–Valtr in 2003. In this paper, we answer both these questions for every tree F. We give a forbidden subgraph characterization for a family $${\mathcal {T}}$$ of ordered trees with k edges, and show that $$\mathrm{{ex}}_{\rightarrow }(n,T) = (k - 1)n - {k \atopwithdelims ()2}$$ for all $$n \ge k + 1$$ when $$T \in {{\mathcal {T}}}$$ and $$\mathrm{{ex}}_{\rightarrow }(n,T) = \Omega (n\log n)$$ for $$T ot \in {{\mathcal {T}}}$$ . We also describe the family $${{\mathcal {T}}}'$$ of the convex geometric trees with linear Turan number and show that for every convex geometric tree $$F otin {{\mathcal {T}}}'$$ , $$\mathrm{{ex}}_{\circlearrowright }(n,F)= \Omega (n\log \log n)$$ .

On Separating Points by Lines

Abstract: Given a set $$P$$ of n points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other, denoted by $$\mathrm {sep}\left( {P}\right) $$. We show that the minimum number of lines needed to separate n points, picked randomly (and uniformly) in the unit square, is $$\Bigl .{\widetilde{\Theta }}( n^{2/3})$$, where $${\widetilde{\Theta }}$$ hides polylogarithmic factors. In addition, we provide a fast $$O(\log (\mathrm {sep}\left( {P}\right) ))$$-approximation algorithm for computing the separability of a given point set in the plane. Finally, we point out the connection between separability and partitions.

Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

Abstract: Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge–Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency. In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.

Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs

Abstract: We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs We first propose an exact (and deterministic) algorithm which solves the problem in $$O(n\log ^2\!n)$$ time using linear space, where n is the number of the vertices of the graph This significantly improves the previous deterministic algorithm by Cabello and Jejcic [CGTA’15] which uses $$O(n^{1+\delta })$$ time and $$O(n^{1+\delta })$$ space (for any constant $$\delta >0$$ ) and the previous randomized algorithm by Kaplan et al [SODA’17] which uses $$O(n\log ^{12+o(1)}\!n)$$ expected time and $$O(n\log ^3\!n)$$ space More specifically, we show that if the 2D offline insertion-only (additively) weighted nearest-neighbor problem with k operations (ie, insertions and queries) can be solved in f(k) time, then the SSSP problem in weighted unit-disk graphs can be solved in $$O(n\log n+f(n))$$ time Using the same framework with some new ideas, we also obtain a $$(1+\varepsilon )$$ -approximate algorithm for the problem, using $$O(n\log n+n\log ^2(1/\varepsilon ))$$ time and linear space This improves the previous $$(1+\varepsilon )$$ -approximate algorithm by Chan and Skrepetos [SoCG’18] which uses $$O((1/\varepsilon )^2n\log n)$$ time and $$O((1/\varepsilon )^2 n)$$ space More specifically, we show that if the 2D offline insertion-only weighted nearest-neighbor problem with $$k_1$$ operations in which at most $$k_2$$ operations are insertions can be solved in $$f(k_1,k_2)$$ time, then the $$(1+\varepsilon )$$ -approximate SSSP problem in weighted unit-disk graphs can be solved in $$O(n\log n+f(n,O(\varepsilon ^{-2})))$$ time Because of the $$\Omega (n\log n)$$ -time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal

Coloring Intersection Hypergraphs of Pseudo-Disks

Abstract: We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with four colors and a conflict-free coloring with $$O(\log n)$$ colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with $$O(\log n)$$ colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.

Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

Abstract: We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, $$Q^*$$-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

Barycentric Subdivisions of Convex Complexes are Collapsible

Abstract: A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for $$d \le 3$$ . We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in $$\mathbb {R}^d$$ is simplicially collapsible after $$d-2$$ derived subdivisions at most. This presents progress on an old question by Goodrick.

Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

Abstract: The covering type of a space $X$ is a numerical homotopy invariant which in some sense measures the homotopical size of $X$. It was first introduced by Karoubi and Weibel (in Enseign Math 62(3-4):457-474, 2016) as the minimal cardinality of a good cover of a space $Y$ taken among all spaces that are homotopy equivalent to $X$. We give several estimates of the covering type in terms of other homotopy invariants of $X$, most notably the ranks of the homology groups of $X$, the multiplicative structure of the cohomology ring of $X$ and the Lusternik-Schnirelmann category of $X$. In addition, we relate the covering type of a triangulable space to the number of vertices in its minimal triangulations. In this way we derive within a unified framework several estimates of vertex-minimal triangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments.

On the Volume of the John–Löwner Ellipsoid

Abstract: We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the Lowner ellipsoid of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace, which are respectively $$\bigl (\frac{n}{k}\bigr )^{{k}/{2}}$$ and $$\bigl (\frac{k}{n}\bigr )^{{k}/{2}}$$ of the volume of the unit ball in $$\mathbb {R}^k$$. Also, we describe all possible vectors in $$\mathbb {R}^n,$$ whose coordinates are the squared lengths of a projection of the standard basis in $$\mathbb {R}^n$$ onto a k-dimensional subspace.

Conflict-Free Coloring of Intersection Graphs of Geometric Objects

Abstract: In 2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-free colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with $$O(\log n)$$ colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family $$\mathcal {F}$$ of n discs in the plane can be colored with $$O(\log n)$$ colors in such a way that for any disc B in the plane, not necessarily from $$\mathcal {F}$$ , the set of discs in $$\mathcal {F}$$ that intersect B contains a uniquely-colored element. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of $$\rho $$ -fat objects in the plane.

Bisectors and Pinned Distances

Abstract: We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their bisector energy.

Decomposing Arrangements of Hyperplanes: VC-Dimension, Combinatorial Dimension, and Point Location

Abstract: This work is motivated by several basic problems and techniques that rely on space decomposition of arrangements of hyperplanes in high-dimensional spaces, most notably Meiser’s 1993 algorithm (Meiser in Inf Comput 106(2):286–303, 1993) for point location in such arrangements. A standard approach to these problems is via random sampling, in which one draws a random sample of the hyperplanes, constructs a suitable decomposition of its arrangement, and recurses within each cell of the decomposition with the subset of hyperplanes that cross the cell. The efficiency of the resulting algorithm depends on the quality of the sample, which is controlled by various parameters. One of these parameters is the classical VC-dimension, and its associated primal shatter dimension, of a suitably defined corresponding range space. Another parameter, which we refer to here as the combinatorial dimension, is the maximum number of hyperplanes that are needed to define a cell that can arise in the decomposition of some sample of the input hyperplanes; this parameter arises in Clarkson’s (and later Clarkson and Shor’s) random sampling technique. We re-examine these parameters for the two main space decomposition techniques—bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension d, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. Our main application is to point location in an arrangement of n hyperplanes is $$\mathbb {R}^d$$, in which we show that the query cost in Meiser’s algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage (which seem to be stated incorrectly, and are not worked out, in Meiser’s work). Our improved bounds rely on establishing several new structural properties and improved complexity bounds for vertical decomposition, which are of independent interest, and which we expect to find additional applications.

Simple Realizability of Complete Abstract Topological Graphs Simplified

Abstract: An abstract topological graph (briefly an AT-graph) is a pair $$A=(G,{\mathcal {X}})$$ where $$G=(V,E)$$ is a graph and $${\mathcal {X}}\subseteq {E \atopwithdelims ()2}$$ is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from $${\mathcal {X}}$$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $${\mathbb {Z}}_2$$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.

A Crossing Lemma for Multigraphs

Abstract: Let G be a drawing of a graph with n vertices and $$e>4n$$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvatal, Newborn, Szemeredi and Leighton, the number of crossings in G is at least $$c\,{e^3\over n^2}$$ , for a suitable constant $$c>0$$ . In a seminal paper, Szekely generalized this result to multigraphs, establishing the lower bound $$c\,{e^3\over mn^2}$$ , where m denotes the maximum multiplicity of an edge in G. We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least $$c'{e^3\over n^2}$$ for some $$c'>0$$ , provided that the “lens” enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.

The Schläfli Fan

Abstract: Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $$344\, 843 \,867$$ such cones, organized into a database of $$14\,373\,645$$ symmetry classes. The Schlafli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.

An Exploration of Locally Spherical Regular Hypertopes

Abstract: Hypertope is a generalization of the concept of polytope, which in turn generalizes the concept of a map and hypermap, to higher rank objects. Regular hypertopes with spherical residues, which we call regular locally spherical hypertopes, must be either of spherical, euclidean, or hyperbolic type. That is, the type-preserving automorphism group of a locally spherical regular hypertope is a quotient of a finite irreducible, infinite irreducible, or compact hyperbolic Coxeter group. We classify the locally spherical regular hypertopes of spherical and euclidean type and investigate finite hypertopes of hyperbolic type, giving new examples and summarizing some known results.

Constructing Planar Support for Non-Piercing Regions

Abstract: Given a hypergraph $$\mathcal {H}=(X,{\mathcal {S}})$$ , a planar support for $$\mathcal {H}$$ is a planar graph G with vertex set X, such that for each hyperedge $$S\in \mathcal {S}$$ , the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph $${\mathcal {H}}_R(B)=(B,\{B_{r}\}_{r\in R})$$ , where $$B_r=\{b\in B:b\cap r e \emptyset \}$$ has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in $$R\cup B$$ . Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTAS’s for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

Tri-partitions and Bases of an Ordered Complex

Abstract: Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.