Computing Irreducible Decomposition of Monomial Ideals

TL;DR

This paper introduces two algorithms for irreducible decomposition of monomial ideals, with the incremental algorithm being more efficient for generic cases and having bounded intermediate storage.

Contribution

It proposes a new incremental algorithm for monomial ideal decomposition with improved efficiency and storage bounds for generic ideals.

Findings

The incremental algorithm outperforms the recursive one for generic monomial ideals.

The time complexity of the incremental algorithm is at most O(n^2 p ℓ).

Intermediate storage in the incremental algorithm is bounded by the output size for generic ideals.

Abstract

The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that the second algorithm is more efficient than the first one for generic monomial ideals. Furthermore, the time complexity of the second algorithm is at most $O(n^2p\ell)$ where $n$ is the number of variables, $p$ is the number of minimal generators and $\ell$ is the number of irreducible components. Another novelty of the second algorithm is that, for generic monomial ideals, the intermediate storage is always bounded by the final output size which may be exponential in the input size.