Greedy Sets and Greedy Numerical Semigroups

TL;DR

This work extends the change-making notion of greediness to sets of positive integers that do not contain $1$ and to numerical semigroups. It introduces $\mathrm{MinRep}_S(k)$, $\mathrm{MinCost}_S(k)$ and a generalized greedy representation $\mathrm{GenGreedyRep}_S(k)$, and develops a witness-based framework to decide greediness via a critical range bound. The authors prove that all two-generator sets are greedy and analyze embedding-dimension-three cases, showing that $\langle n,n+1,n+2\rangle$ is greedy, while providing a practical algorithm to certify greediness through witnesses. They also establish complexity boundaries: the witness-search approach is polynomial in the number of generators and their largest value, but related problems are NP-hard or co-NP-complete, highlighting both practical methods and fundamental limitations. The paper closes with open questions and directions for future work, including identifying infinite greedy families and understanding greediness under semigroup operations and invariants.

Abstract

Motivated by the change-making problem, we extend the notion of greediness to sets of positive integers not containing the element $1$, and from there to numerical semigroups. We provide an algorithm to determine if a given set (not necessarily containing the number $1$) is greedy. We also give specific conditions for sets of cardinality three, and we prove that numerical semigroups generated by three consecutive integers are greedy.

Theorems & Definitions (20)

• Definition 1
• Proposition 1: Triangle inequality
• Definition 2
• Lemma 1
• Proposition 2
• Example 1
• Definition 3
• Example 2
• Lemma 2
• Definition 4