Helly Theorems Research Articles

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter $$d\ge 1$$ , in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For $$d=1$$ , this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor $$n^{1-\epsilon }$$ , for any $$\epsilon >0$$ . Moreover, it is known that, for any $$d\ge 2$$ , it is NP-hard to approximate MaxDBS within a factor $$n^{1/2-\epsilon }$$ , for any $$\epsilon >0$$ . In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in Rd has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a “witness set” which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and H-convex sets. Our results show that several new optimization problems can be solved with algorithms for LP-type problems. We obtain colorful and fractional variants of all our Helly-type theorems.

The affine separation problem revisited

The aim of this note is to characterize in terms of inequalities those pairs of real functions (acting on a convex subset of a vector space) that possess an affine separator. The main result is originally due to Behrends and Nikodem. Their method is based on the Hahn–Banach Theorem and a variant of the Helly Theorem. In our approach, a direct and independent proof is presented via the Radon and the Helly Theorems.

Bayesian Model Selection: Measuring the χ2 Discrepancy with the Uniform Distribution

In the last years, many articles have been written about Bayesian model selection. In this article, a different and easier method is proposed and analyzed. The key idea of this method is based on the well-known property that, under the true model, the cumulative distribution function is distributed as a uniform distribution over the interval (0, 1). The method is first introduced for the continuous case and then for the discrete case by smoothing the cumulative distribution function. Some asymptotical properties of the method are obtained by developing an alternative to Helly's theorems. Finally, the performance of the method is evaluated by simulation, showing a good behavior.

On the Algorithmic Aspects of Discrete and Lexicographic Helly-Type Theorems and the Discrete LP-Type Model

Helly's theorem says that, if every $d+1$ elements of a given finite set of convex objects in $\mathbb{R}^d$ have a common point, there is a point common to all of the objects in the set. In discrete Helly theorems the common point should belong to an a priori given set. In lexicographic Helly theorems the common point should not be lexicographically greater than a given point. Using discrete and lexicographic Helly theorems we get linear time solutions for various optimization problems. For this, we introduce the DLP-type (discrete linear programming-type) model, and provide new algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. For variable-dimensional DLP-type problems, our algorithms run in time subexponential in the combinatorial dimension. Finally, we use our results in order to solve in randomized linear time problems such as the discrete $p$-center on the real line, the discrete weighted 1-center problem in $\mathbb{R}^d$ with either $l_1$ or $l_\infty$ norm, the standard (continuous) problem of finding a line transversal for a totally separable set of planar convex objects, a discrete version of the problem of finding a line transversal for a set of axis-parallel planar rectangles, and the (planar) lexicographic rectilinear $p$-center problem for $p=1,2,3$. These are the first known linear time algorithms for these problems. Moreover, we use our algorithms to solve in randomized subexponential time various problems in game theory, improving upon the best known algorithms for these problems.

Discrete and Lexicographic Helly-Type Theorems

Helly’s theorem says that if every d+1 elements of a given finite set of convex objects in ℝd have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems—where the common point should belong to an a-priori given set, lexicographic Helly theorems—where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers.

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

Cell complexes, oriented matroids and digital geometry

Abstract cell complexes (ACCs) were introduced by Kovalevsky as a means of solving certain connectivity paradoxes in graph-theoretic digital topology, and to this extent provide an improved theoretical basis for image analysis. In this work we argue that ACCs are a very natural setting for digital geometry, to the extent that their use permits simple, almost trivial formulations of major convexity results, including Caratheodory's, Helly's and Radon's theorems. We also discuss the relevance of oriented matroids to digital geometry.

A Proof of the Equivalence of Helly's and Krasnoselski's Theorems

AbstractWe show that Krasnoselski's Theorem, which is usually derived from Helly's Theorem, is in fact equivalent to it.

Some Helly Theorems for Monotone Functions

Some Helly Theorems for Monotone Functions

Some Helly theorems for monotone functions

Some Helly theorems for monotone functions