A heuristic technique for decomposing multisets of non-negative integers according to the Minkowski sum

TL;DR

This paper introduces a heuristic method for decomposing multisets of non-negative integers based on Minkowski sums, enabling faster polynomial factorization in N[x] compared to existing algorithms.

Contribution

The paper presents a novel heuristic technique for multiset decomposition related to Minkowski sums, improving speed and scalability over current methods.

Findings

Decomposes multisets of hundreds of elements within seconds.

Faster than existing algorithms for polynomial factorization in N[x].

Effective regardless of the magnitude of multiset elements.

Abstract

We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.