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

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 betwee...

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Bibliographic Details
Published inJournal of optimization theory and applications Vol. 187; no. 2; pp. 431 - 447
Main Author Runge, Vincent
Format Journal Article
LanguageEnglish
Published New York Springer US 01.11.2020
Springer Nature B.V
Subjects
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Computational geometry
Convexity
Dynamic programming
Engineering
Enumeration
Euclidean geometry
Euclidean space
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Polyhedra
Polynomials
Programming languages
Quadratic programming
Theory of Computation
Computational geometry
Nonconvex quadratically constrained quadratic programming
Voronoi-like polyhedron
Vertex enumeration
90C26
Polynomial time complexity
68U05
52C17
Ball covering problem
62L10
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ISSN0022-3239
1573-2878
DOI10.1007/s10957-020-01762-2

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Summary: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-020-01762-2