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：本文研究在球交集中最大化到给定点的距离问题。已知当给定点不在球心凸包中时，该问题可在多项式时间与空间内求解。我们证明当给定点位于球心凸包内时，该问题涵盖所有NP完全问题，并在此基础上给出若干新结果。我们提出一种新的投影算法，并将其应用于子集和问题(SSP)中。在假设SSP至多只有一个解的前提下，我们设计了一种拟多项式算法，通过缩小包含SSP解的初始球半径来逼近最优解。数值实验表明所提算法具有有效性。