Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces ?
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Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we build an exact and efficient test answering the question of the cover of the intersection by the union. This covering problem can be reformulated into quadratic programming problems, whose resolution for minimum and maximum gives information about a possible overlap between the frontier of the union and the intersection of balls. Obtained feasible regions are convex polyhedra, which are non-degenerate for many applications. Therefore, the initial nonconvex geometric problem, which is NP-hard in general, is now tractable in polynomial time by vertex enumeration. This time complexity reduction is due to the simple geometry of our problem involving only balls. The nonconvex maximum problems can be skipped when some mild conditions are satisfied. In this case, we only solve a collection of convex quadratic programming problems in polynomial time complexity. Simulations highlight the accuracy and efficiency of our approach compared with competing algorithms 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 for genomic data.
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