In this paper we study the Subset Sum Problem (SSP). Assuming the SSP has at most one solution, we provide a randomized quasi-polynomial algorithm which if the SSP has no solution, the algorithm always returns FALSE while if the SSP has a solution the algorithm returns TRUE with probability $\frac{1}{2^{\log(n)}}$. This can be seen as two types of coins. One coin, when tossed always returns TAILS while the other also returns HEADS but with probability $\frac{1}{2^{\log(n)}}$. Using the Law of Large Numbers one can identify the coin type and as such assert the existence of a solution to the SSP. The algorithm is developed in the more general framework of maximizing the distance to a given point over an intersection of balls.

翻译：本文研究子集和问题（SSP）。假设SSP至多有一个解，我们提出一种随机拟多项式算法：若SSP无解，算法始终返回FALSE；若SSP有解，算法以概率$\frac{1}{2^{\log(n)}}$返回TRUE。这可以视为两类硬币：一类硬币抛掷时始终返回反面，另一类硬币以概率$\frac{1}{2^{\log(n)}}$返回正面。利用大数定律可识别硬币类型，从而断言SSP解的存在性。该算法是在更一般的框架——在球体交集上最大化到给定点的距离——中开发的。