Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smallest Intersecting and Enclosing Balls

Jiaqi Zheng, Tiow-Seng Tan

TL;DR

The paper studies the smallest intersecting ball ($SIB$) and smallest enclosing ball ($SEB$) problems for compact convex objects in $\\mathbb{R}^d$, introducing a unifying zero-sum game framework. It demonstrates that $SIB$ is bilinear while $SEB$ is not, and proves the existence of Nash equilibria whose values equal $r^*$ and $R^*$ respectively. Leveraging the bilinear structure, the authors develop the first $(1+\varepsilon)$-approximation algorithm for $SIB$ in arbitrary dimensions, with a running time of $O\left(\dfrac{R^2 (S + nd) \log n}{\varepsilon^2}\right)$ given a best-response cost $O(S)$. This work advances high-dimensional geometric optimization and provides a practical method for computing tight intersecting balls across diverse convex inputs, with broader implications for related problems in computational geometry and machine learning.

Abstract

We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems can be modeled as zero-sum games, and propose an approximation algorithm for the former. Specifically, the algorithm produces the first results in high-dimensional spaces for various input objects such as convex polytopes, balls, ellipsoids, etc.

Smallest Intersecting and Enclosing Balls

TL;DR

The paper studies the smallest intersecting ball () and smallest enclosing ball () problems for compact convex objects in , introducing a unifying zero-sum game framework. It demonstrates that is bilinear while is not, and proves the existence of Nash equilibria whose values equal and respectively. Leveraging the bilinear structure, the authors develop the first -approximation algorithm for in arbitrary dimensions, with a running time of given a best-response cost . This work advances high-dimensional geometric optimization and provides a practical method for computing tight intersecting balls across diverse convex inputs, with broader implications for related problems in computational geometry and machine learning.

Abstract

We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems can be modeled as zero-sum games, and propose an approximation algorithm for the former. Specifically, the algorithm produces the first results in high-dimensional spaces for various input objects such as convex polytopes, balls, ellipsoids, etc.

Paper Structure

This paper contains 3 sections, 3 theorems, 3 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. SIB and SEB as Zero-Sum Games
  3. Algorithms

Key Result

Theorem 1

The SIB problem can be modeled as the following zero-sum game: A Nash equilibrium (denoted as $({\boldsymbol{x}}^*, {\boldsymbol{v}}_1^*, \dots, {\boldsymbol{v}}_n^*, {\boldsymbol{y}}^*)$) of the game always exist, and the value of the game is $r^*$. The ball $B({\boldsymbol{x}}^*, r^*)$ is intersecting every $\Omega_i$, and ${\boldsymbol{v}}_i^* \in B({\boldsy

Figures (2)

  • Figure 1: Examples of the problems in 2D spaces, where the blue objects are the input and red circles are the solutions. Left: the smallest intersecting ball. Right: the smallest enclosing ball.
  • Figure 2: Many faces of the SIB problem. Left: the SEB of a point set. Middle: the nearest point (Euclidean projection) in a convex set. Right: the shortest connector (minimum distance).

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3