In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of the balls centers. The cases where the given point is in the convex hull of the balls centers include all NP-complete problems as we show. Some novel results are given in this area. A novel projection algorithm is developed then applied in the context of the Subset Sum Problem (SSP). Under the assumption that the SSP has at most one solution, we provide a quasi-polynomial algorithm, which decreases the radius of an initial ball containing the solution to the SSP. We perform some numerical tests which show the effectiveness of the proposed algorithm.

翻译：本文研究了在球交集上最大化到给定点的距离的问题。已知若给定点不在球心凸包内，则可在多项式时间和空间内解决该问题。我们证明，给定点位于球心凸包内的情况包含了所有NP完全问题。本文在该领域给出了一些新颖成果，并开发了一种新型投影算法，随后将其应用于子集和问题（SSP）。在假设SSP至多有一个解的条件下，我们提出了一种拟多项式算法，该算法能缩小包含SSP解的初始球半径。数值实验表明该算法的有效性。