  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.
