[Paper Review] The Computational Complexity of the Frobenius Problem

Shunichi Matsubara|arXiv (Cornell University)|Feb 18, 2016

Commutative Algebra and Its Applications28 references3 citations

TL;DR

This paper establishes the computational complexity of the decision version of the Frobenius problem by proving it is $Σ_2^P$-complete under Karp reductions. The proof involves two Karp reductions: from a $\Pi_2^P$-complete variant of 3D matching to a $\Pi_2^P$-complete variant of the integer knapsack problem, and then to the complement of the Frobenius decision problem, thereby establishing the first nontrivial upper and lower bounds for the problem's complexity.

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.

Motivation & Objective

• To determine the computational complexity of the decision version of the Frobenius problem, which asks whether the Frobenius number $g(A)$ of a set $A$ of coprime integers is at least a given $k$.
• To resolve an open question posed by Ramírez-Alfonsín regarding whether the Frobenius problem is $Σ_2^P$-complete under Karp reductions.
• To provide the first nontrivial upper and lower bounds for the function version of the Frobenius problem, $F$ Frobenius, by establishing its complexity relative to the polynomial hierarchy.
• To analyze the problem's structure through reductions from known $\Pi_2^P$-complete problems, particularly a $2$-alternating variant of 3D matching and a $2$-alternating integer knapsack problem.

Proposed method

• Construct a Karp reduction from the $\Pi_2^P$-complete problem $\Pi_2$3D Matching to a $\Pi_2^P$-complete variant of the integer knapsack problem, termed $\Pi_2$Assoc Integer Knapsack.
• Define $\Pi_2$Assoc Integer Knapsack as a $2$-alternating version of the integer knapsack problem, which captures the logical alternation of existential and universal quantifiers relevant to the Frobenius problem.
• Establish a Karp reduction from $\Pi_2$Assoc Integer Knapsack to the complement of the decision Frobenius problem, $\text{coFrobenius}$, showing $\text{coFrobenius}$ is $\Pi_2^P$-hard.
• Prove that $\text{coFrobenius}$ is in $\Pi_2^P$, thereby establishing $\text{coFrobenius} \in \Pi_2^P$ and $\text{coFrobenius}$ is $\Pi_2^P$-complete.
• Derive the $\Sigma_2^P$-completeness of the decision Frobenius problem via duality, since $\text{Frobenius} = \neg \text{coFrobenius}$, and use this to bound the complexity of the function version $F$ Frobenius.
• Analyze the algorithmic implications by showing that the Frobenius problem admits a pseudopolynomial-time algorithm, implying it is weakly $\Sigma_2^P$-hard.

Experimental results

Research questions

• RQ1 Is the decision version of the Frobenius problem $\Sigma_2^P$-complete under Karp reductions?
• RQ2 Can the Frobenius problem be reduced to or from known $\Pi_2^P$-complete problems, such as $\Pi_2$3D Matching or $\Pi_2$Assoc Integer Knapsack?
• RQ3 What is the tightest possible complexity class for the function version of the Frobenius problem, $F$ Frobenius, given the decision version's complexity?
• RQ4 Does the existence of a pseudopolynomial-time algorithm for the Frobenius problem imply that it is not strongly $\Sigma_2^P$-hard?
• RQ5 Can the Frobenius problem serve as a candidate to resolve major open questions in complexity theory, such as $\mathsf{NP} = \Sigma_2^P$?

Key findings

• The decision version of the Frobenius problem, Frobenius, is proven to be $\Sigma_2^P$-complete under Karp reductions, resolving an open problem posed by Ramírez-Alfonsín.
• The complement problem, $\text{coFrobenius}$, is shown to be $\Pi_2^P$-complete via a Karp reduction from $\Pi_2$Assoc Integer Knapsack, a newly defined $2$-alternating variant of the integer knapsack problem.
• The function version of the Frobenius problem, $F$ Frobenius, is weakly $\Sigma_2^P$-hard, as demonstrated by a pseudopolynomial-time algorithm based on shortest path computation over a weighted directed graph derived from the input set $A$.
• The problem is shown to be in $\mathsf{D}_2^P$, and the paper conjectures that Exact Frobenius may be $\mathsf{D}_2^P$-complete, though this remains open.
• The existence of a pseudopolynomial algorithm implies that the problem is not strongly $\Sigma_2^P$-hard, suggesting potential for approximation schemes, though no FPTAS is currently known.
• The result provides the first nontrivial upper bound for $F$ Frobenius, placing it within $\mathsf{F}\Delta_3^P$, and opens avenues for future research on approximation and structural complexity.