On the Fine-Grained Complexity of the Unbounded SubsetSum and the Frobenius Problem
share
Consider positive integral solutions x ∈ℤ^n+1 to the equation a_0 x_0 + … + 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.
READ FULL TEXT