Computing all minimal Markov bases in Macaulay2

TL;DR

The paper addresses the challenge of enumerating all minimal Markov bases of a toric ideal by introducing the allMarkovBases package for Macaulay2, which employs fiber graphs $G_t$ and an all-Markov algorithm to systematically construct the bases from generating fibers $t \\in \\mathbb{N}A$. It provides practical tools to compute the indispensable set $S(A)$ and the universal Markov basis $U(A)$ directly from fiber graphs, and supports uniform random sampling when the total number of bases is very large. Key contributions include an explicit algorithmic framework based on Prüfer sequence–driven spanning-tree enumeration, demonstrations on small and large examples, and efficient storage of fiber graphs to avoid recomputation. The approach enhances reproducible, scalable analysis of Markov bases in algebraic statistics, enabling both complete enumeration and probabilistic sampling for complex configurations.

Abstract

We introduce the package allMarkovBases for Macaulay2, which is used to compute all minimal Markov bases of a given toric ideal. The package builds on functionality of 4ti2 by producing the fiber graph of the toric ideal. The package uses this graph to compute other properties of the toric ideal such as its indispensable set of binomials as well as its universal Markov basis.

Figures (2)

• Figure 1: Fiber graphs of $A$ for the generating fibers
• Figure 2: Spanning tree on $\{0,1,\dots,5\}$ associated to the Prüfer sequence $\{0,0,2,4\}$

Theorems & Definitions (1)

• Remark 2.1