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.

中文翻译：

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球的有限交集是否被欧几里德空间中的球的有限联合所覆盖？

考虑到欧几里德空间中球的有限交集和其他球的有限并集，我们构建了一个准确有效的测试来回答并集覆盖交集的问题。这个覆盖问题可以重新表述为二次规划问题，其最小值和最大值的分辨率给出了关于联合边界和球交集之间可能重叠的信息。获得的可行区域是凸多面体，它在许多应用中都是非退化的。因此，最初的非凸几何问题通常是 NP 难的，现在可以通过顶点枚举在多项式时间内处理。这种时间复杂度的降低是由于我们问题的简单几何结构只涉及球。当满足一些温和的条件时，可以跳过非凸极大值问题。在这种情况下，我们只解决多项式时间复杂度中的一组凸二次规划问题。与非凸二次约束二次规划的竞争算法相比，模拟突出了我们方法的准确性和效率。这项工作的动机是在统计学中使用修剪的动态编程算法对基因组数据进行多维变化点检测问题的应用。