On the Complexity of Closest Pair via Polar-Pair of Point-Sets

TL;DR

This paper explores the geometric representation of graphs using spheres in various metrics, focusing on the sphericity and contact dimension of complete bipartite graphs, and links these concepts to the complexity of closest pair problems.

Contribution

It introduces the concepts of sphericity and contact dimension for graph representations and analyzes these parameters for complete bipartite graphs in different $L^p$-metrics, connecting to closest pair problem complexities.

Findings

Determines sphericity and contact dimension of $K_{n,n}$ in various $L^p$-metrics.

Establishes a connection between graph geometric representations and closest pair problem complexities.

Provides bounds and characterizations for sphere representations of bipartite graphs.

Abstract

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.