Consider positive integral solutions $x \in \mathbb{Z}^{n+1}$ to the equation $a_0 x_0 + \ldots + a_n x_n = t$. In the so called unbounded subset sum problem, the objective is to decide whether such a solution exists, whereas in the Frobenius problem, the objective is to compute the largest $t$ such that there is no such solution. In this paper we study the algorithmic complexity of the unbounded subset sum, the Frobenius problem and a generalization of the problems. More precisely, we study pseudo-polynomial time algorithms with a running time that depends on the smallest number $a_0$ or respectively the largest number $a_n$. For the parameter $a_0$, we show that all considered problems are subquadratically equivalent to $(min,+)$-convolution, a fundamental algorithmic problem from the area of fine-grained complexity. By this equivalence, we obtain hardness results for the considered problems (based on the assumption that an algorithm with a subquadratic running time for $(min,+)$-convolution does not exist) as well as algorithms with improved running time. The proof for the equivalence makes use of structural properties of solutions, a technique that was developed in the area of integer programming. In case of the complexity of the problems parameterized by $a_n$, we present improved algorithms. For example we give a quasi linear time algorithm for the Frobenius problem as well as a hardness result based on the strong exponential time hypothesis.

翻译：考虑正整体解决方案 $x $x a_ 0 x_0 +\ ldots + a_n x_n = t$ 。 在所谓的未受约束子子集问题中,目标是确定是否存在这样的解决方案,而在Frobenius问题中,目标是计算最大的美元,这样就不存在这样的解决方案。在本文中,我们研究了未受约束子集的算法复杂性、 Frobenius 问题和问题的一般化。更确切地说,我们研究假的 polynomiaal 时间算法,其运行时间取决于最小的 $_ 0 $ 或最大的子组和 $ 。 关于参数 $_ 0, 我们发现所有考虑的问题都亚基值相当于$( min, +) 递增。 在微缩缩缩缩略图中, 我们通过这个等等值, 我们从所考虑的解决方案中获得了坚硬的结果( 依据假设的是, 以最小值 美元 递增的算法 将 的 时间区域作为我们所发展到的变变变变的变的 。