On the Fine-Grained Complexity of the Unbounded SubsetSum and the Frobenius Problem

TL;DR

This paper explores the fine-grained computational complexity of the unbounded subset sum and Frobenius problems, establishing equivalences with the min-plus convolution and providing both hardness results and improved algorithms.

Contribution

It introduces a detailed complexity analysis, showing equivalences with min-plus convolution and developing faster algorithms for specific parameter regimes.

Findings

Problems are subquadratically equivalent to min-plus convolution for parameter a0.

Established hardness results based on the assumption of no subquadratic min-plus algorithms.

Presented a quasi-linear time algorithm for the Frobenius problem when parameterized by a_n.

Abstract

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.