The Unbounded Subset Sum (USS) problem is an NP-hard computational problem where the goal is to decide whether there exist non-negative integers $x_1, \ldots, x_n$ such that $x_1 a_1 + \ldots + x_n a_n = b$, where $a_1 < \cdots < a_n < b$ are distinct positive integers with $\text{gcd}(a_1, \ldots, a_n)$ dividing $b$. The problem can be solved in pseudopolynomial time, while specialized cases, such as when $b$ exceeds the Frobenius number of $a_1, \ldots, a_n$ simplify to a total problem where a solution always exists. This paper explores the concept of totality in USS. The challenge in this setting is to actually find a solution, even though we know its existence is guaranteed. We focus on the instances of USS where solutions are guaranteed for large $b$. We show that when $b$ is slightly greater than the Frobenius number, we can find the solution to USS in polynomial time. We then show how our results extend to Integer Programming with Equalities (ILPE), highlighting conditions under which ILPE becomes total. We investigate the diagonal Frobenius number, which is the appropriate generalization of the Frobenius number to this context. In this setting, we give a polynomial-time algorithm to find a solution of ILPE. The bound obtained from our algorithmic procedure for finding a solution almost matches the recent existential bound of Bach, Eisenbrand, Rothvoss, and Weismantel (2024).

翻译：无界子集和（USS）问题是一个NP难计算问题，其目标是判断是否存在非负整数 $x_1, \ldots, x_n$，使得 $x_1 a_1 + \ldots + x_n a_n = b$，其中 $a_1 < \cdots < a_n < b$ 是互异正整数且满足 $\text{gcd}(a_1, \ldots, a_n)$ 整除 $b$。该问题可在伪多项式时间内求解，而某些特殊情形（例如当 $b$ 超过 $a_1, \ldots, a_n$ 的弗罗贝尼乌斯数时）会简化为解必然存在的“全问题”。本文探讨USS中的全性问题。在此设定下的挑战在于实际找出一个解，尽管我们已知解的存在性是有保证的。我们聚焦于解在 $b$ 较大时必然存在的USS实例。我们证明，当 $b$ 略大于弗罗贝尼乌斯数时，可以在多项式时间内找到USS的解。随后，我们展示了如何将结果推广至等式型整数规划（ILPE），并阐明了ILPE成为全问题的条件。我们研究了对角弗罗贝尼乌斯数——这是弗罗贝尼乌斯数在此语境下的恰当推广。在此框架下，我们给出了一种在多项式时间内求解ILPE的算法。我们算法求解过程所得到的界，几乎匹配了Bach、Eisenbrand、Rothvoss与Weismantel（2024）近期提出的存在性界。