The Complexity of the Numerical Semigroup Gap Counting Problem

TL;DR

This paper establishes that counting gaps in numerical semigroups is a #NP-complete problem, resolving a long-standing question about its computational complexity and demonstrating its difficulty.

Contribution

It proves that the numerical-semigroup-gap counting problem is #NP-complete, answering a question posed in 2005 and extending complexity results to related variants.

Findings

Numerical-semigroup-gap counting problem is #NP-complete.

Two variants of the problem are also #NP-complete.

The work resolves a long-standing open question in computational complexity.

Abstract

In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined as a positive integer that does not belong to the numerical semigroup. The computation of gaps of numerical semigroups has been actively studied from the 19th century. However, little has been known on the computational complexity. In 2005, Ramirez-Alfonsin proposed a question whether or not the numerical-semigroup-gap counting problem is #P-complete. This work is an answer for his question. For proving the main theorem, we show the #NP-completenesses of other two variants of the numerical-semigroup-gap counting problem.