The Computational Complexity of the Frobenius Problem

Abstract: In this paper, as a main theorem, we prove that the decision version of the Frobenius problem is Sigma_2^P-complete under Karp reductions.Given a finite set A of coprime positive integers, we call the greatest integer that cannot be represented as a nonnegative integer combination of A the Frobenius number, and we denote it as g(A). We call a problem of finding g(A) for a given A the Frobenius problem; moreover, we call a problem of determining whether g(A) >= k for a given pair (A, k) the decision version of the Frobenius problem, where A is a finite set of coprime positive integers and k is a positive integer. For the proof, we construct two Karp reductions. First, we reduce a 2-alternating version of the 3-dimensional matching problem, which is known to be Pi_2^P-complete, to a 2-alternating version of the integer knapsack problem. Then, we reduce the variant of the integer knapsack problem to the complement of the decision version of the Frobenius problem. As a corollary, we obtain the main theorem.