# Discrete & Computational Geometry

## List of Papers (Total 74)

#### Coloring $d$ -Embeddable $k$ -Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. Let $\fancyscript{H}_{d,k}$ be the set of all $k$-uniform hypergraphs that can be (linearly) embedded into $\mathbb {R}^d$. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in $\fancyscript{H}_{d,k}$. For example, we can prove that for $d\ge 3$ there ...

#### Coloring Intersection Graphs of Arc-Connected Sets in the Plane

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their ...

#### Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R }^2$ that is not an ...

#### $f$ -Vectors Implying Vertex Decomposability

We prove that if a pure simplicial complex $\Delta$ of dimension $d$ with $n$ facets has the least possible number of $(d-1)$-dimensional faces among all complexes with $n$ faces of dimension $d$, then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.

#### The Structure of Cube Tilings Under Symmetry Conditions

Let m 1,…,m d be positive integers, and let G be a subgroup of ℤ d such that m 1ℤ×⋯×m d ℤ⊆G. It is easily seen that if a unit cube tiling [0,1) d +t,t∈T, of ℝ d is invariant under the action of G, then for every t∈T, the number |T∩(t+ℤ d )∩[0,m 1)×⋯×[0,m d )| is divisible by |G|. We give sufficient conditions under which this number is divisible by a multiple of |G|. Moreover, a ...

#### Unions of Fat Convex Polytopes Have Short Skeletons

The skeleton of a polyhedral set is the union of its edges and vertices. Let $\mathcal {P}$ be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in $\mathcal {P}$. We prove that the total length of the skeleton of the union of the polytopes in $\mathcal {P}$ is at most O(α(n)⋅log∗ ...

#### Sphere and Dot Product Representations of Graphs

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 ...

#### A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets (regions or neighborhoods) in the underlying metric space. We give a quasi-polynomial time approximation scheme (QPTAS) when the regions are what we call α-fat weakly disjoint. This notion combines the existing ...

#### Coreduction Homology Algorithm for Regular CW-Complexes

In this paper we present a new algorithm for computing the homology of regular CW-complexes. This algorithm is based on the coreduction algorithm due to Mrozek and Batko and consists essentially of a geometric preprocessing algorithm for the standard chain complex generated by a CW-complex. By employing the concept of S-complexes the original chain complex can—in all known ...

#### Minimum Manhattan Network is NP-Complete

Given a set T of n points in ℝ2, a Manhattan network on T is a graph G with the property that for each pair of points in T, G contains a rectilinear path between them of length equal to their distance in the L 1-metric. The minimum Manhattan network problem is to find a Manhattan network of minimum length, i.e., minimizing the total length of the line segments in the network. In ...

#### Kinetic Spanners in ℝ d

We present a new (1+ε)-spanner for sets of n points in ℝ d . Our spanner has size O(n/ε d−1) and maximum degree O(log d n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n 2/ε d−1), and ...

#### Persistent Cohomology and Circular Coordinates

Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent ...

#### Optimal Topological Simplification of Discrete Functions on Surfaces

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 ...

#### k-means Requires Exponentially Many Iterations Even in the Plane

The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its running time (i.e., n O(kd)) is, in general, exponential in the number of points (when kd=Ω(n/log n)). Recently Arthur and Vassilvitskii ...

#### The Contact Polytope of the Leech Lattice

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.

#### Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle

Given any knot diagram E, we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to E is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in ℝ2.

#### Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.