[Paper Review] Computation of Delta sets of numerical monoids

J. I. García-García, M. A. Moreno-Frías+1 more|arXiv (Cornell University)|Jun 2, 2014

Rings, Modules, and Algebras9 references4 citations

TL;DR

This paper presents an improved algorithm for computing the Delta set of a numerical monoid by introducing a significantly reduced bound $N_S + a_p - 1$ for factorization computation, reducing computational complexity. The method leverages factorization sets and Betti elements to efficiently compute $Δ(S)$, with benchmarks showing up to 99% reduction in search space compared to prior methods, enabling faster and more scalable computation of non-unique factorization invariants.

ABSTRACT

Let $\{a_1,\dots,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $Δ(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1

Motivation & Objective

• To address the computational challenge of determining the Delta set $Δ(S)$ for numerical monoids, which is critical in non-unique factorization theory.
• To refine the existing bound $N$ such that $Δ(S) = igcup_{s < N} Δ(s)$, reducing the search space for factorization computation.
• To develop an efficient algorithm that computes $Δ(S)$ using only the factorizations of $a_1$ elements, minimizing redundant computations.
• To provide a practical, parallelized implementation in Mathematica that outperforms prior methods in execution time and memory usage.

Proposed method

• Derive a new theoretical bound $N_S + a_p - 1$ using Betti elements and factorization structures, improving upon the prior bound $2pa_2a_p^2 + a_1a_p$ from [5].
• Use the structure of factorization sets ${\sf Z}(s)$ and the set of differences $\Omega(s)$ to compute $\Delta(s)$ efficiently for elements up to the new bound.
• Apply a recursive and parallelizable computation strategy to compute factorizations and differences across a bounded range of elements, reducing time complexity.
• Leverage the fact that $\Delta(s) = \Delta(s + a_1a_p)$ for large $s$, allowing truncation of the search space.
• Implement the algorithm in Mathematica using functions like `DeltaSNParallel` to enable practical computation and benchmarking.
• Optimize the algorithm by computing factorizations only for a minimal set of critical elements, reducing the number of NP-hard Diophantine solves.

Experimental results

Research questions

• RQ1 What is the minimal bound $N$ such that $\Delta(S) = \bigcup_{s < N} \Delta(s)$ for a numerical monoid $S$?
• RQ2 Can the bound for computing $\Delta(S)$ be significantly reduced compared to existing results, especially for monoids with large generators?
• RQ3 How can the computation of $\Delta(S)$ be optimized by focusing only on a small number of critical elements?
• RQ4 To what extent does the new bound improve execution time and scalability in practice?
• RQ5 Can the structure of Betti elements and factorization sets be used to derive tighter theoretical bounds for $\Delta(S)$?

Key findings

• The new bound $N_S + a_p - 1$ is drastically smaller than the prior bound $2pa_2a_p^2 + a_1a_p$, with reductions up to 99.9% in tested cases.
• For $S = \langle 15,16,27\rangle$, the new bound is 446, compared to 70,224 in [5], a reduction of over 99%.
• The algorithm computes $\Delta(S)$ using only the factorizations of $a_1$ elements, minimizing computational overhead.
• Execution times are significantly reduced: $\langle 4,6,15\rangle$ computes in 0.018 seconds, while $\langle 7,19,20,25,29\rangle$ takes 878.7 seconds due to larger factorization sets.
• The method scales well for monoids with small $a_1$, and the parallel implementation reduces time for larger monoids like $\langle 31,73,77,87,91\rangle$ to 24,012 seconds.
• The Delta set $\Delta(S)$ is correctly computed for all tested monoids, including complex cases like $\{2,4,6\}$ and $\{1,2,3,5\}$, confirming theoretical correctness.