Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces ?

TL;DR

This paper introduces an exact, polynomial-time method using convex optimization to determine if a finite intersection of balls is covered by a finite union of balls in Euclidean spaces, with applications in statistics.

Contribution

It reformulates the coverage problem into quadratic programming and studies sphere-Voronoi polyhedron intersections, enabling efficient solutions for a class of NP-hard geometric problems.

Findings

Method is accurate and efficient compared to existing algorithms.

Polyhedral approach allows polynomial-time solutions under mild conditions.

Application demonstrated in multidimensional changepoint detection.

Abstract

Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection between a sphere and a Voronoi-like polyhedron. That way we get information about a possible overlap between the frontier of the union and the intersection of balls. If the polyhedra are non-degenerate, the initial nonconvex geometric problem, which is NP-hard in general, is tractable in polynomial time by convex optimization tools and vertex enumeration. Under some mild conditions the vertex enumeration can be skipped. Simulations highlight the accuracy and efficiency of our approach compared with competing algorithms in Python for nonconvex quadratically constrained quadratic programming. This work is motivated by an application in statistics to the problem of multidimensional changepoint detection using pruned dynamic programming algorithms.