Approximation Algorithms for Smallest Intersecting Balls

TL;DR

The paper addresses the general smallest intersecting ball (SIB) problem and its soft-margin variant in high-dimensional spaces for compact convex inputs. It develops a unified framework based on symmetric cone games (SCG) grounded in Euclidean Jordan algebras to obtain $(1+\varepsilon)$-approximation algorithms for both SIB and Soft-SIB, with provable guarantees and parallelizable procedures. The methods apply to diverse input types, including convex polytopes, reduced polytopes, AABBs, balls, and ellipsoids, with efficient subproblem oracles and near-linear scaling in the number of objects $n$ and dimension $d$ in many cases. Experimental results on large-scale instances validate practical efficiency and establish the first high-dimensional approximations for SIB with non-singleton inputs, offering scalable tools for related problems in computational geometry and machine learning.

Abstract

We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces for input objects that are compact and convex. These two problems link and unify a series of fundamental problems in computational geometry and machine learning, including smallest enclosing ball, polytope distance, intersection radius, $\ell_1$-loss support vector machine, $\ell_1$-loss support vector data description, and so on. Leveraging our novel framework for solving zero-sum games over symmetric cones, we propose general approximation algorithms for the two problems, where implementation details are presented for specific inputs of convex polytopes, reduced polytopes, axis-aligned bounding boxes, balls, and ellipsoids. For most of these inputs, our algorithms are the first results in high-dimensional spaces, and also the first approximation methods. Experimental results show that our algorithms can solve large-scale input instances efficiently.

Figures (6)

• Figure 1: Visualization of the 2D results (red circles) computed by our algorithms. The brown and blue lines illustrate the trajectories of the centers and the point in each object respectively.
• Figure 2: Left: the original convex polytope. Middle: the reduced polytopes (with $\nu_1 = \nu_2 = {1\over 2}$), the SIB and the optimal separating hyperplane thereof. Right: the corresponding soft-margin SVM.
• Figure 3: Left: the SIB that intersects the convex hulls of two sets of ellipsoids. Right: the corresponding linear classifier that separates the two sets of ellipsoids with the largest margin.
• Figure 4: Left: the original SIB. Right: the corresponding soft-margin SIB, where $\xi_1 = \xi_2 = \xi_3 = 0$.
• Figure 5: Running time of our algorithm with different input sizes. The shaded regions represent the standard deviations. The number of objects considered differs across object types due to their varying levels of complexity. For each convex polytope and reduced polytope, we fix $m_i = 128$.

Theorems & Definitions (30)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Proposition 4
• proof
• Lemma 1
• proof