Unimodular Equivalence of Integral Simplices

TL;DR

This paper introduces a novel algorithm called HEM for efficiently testing unimodular equivalence of integral simplices, utilizing permuted Hermite normal forms and achieving near-polynomial average-case complexity.

Contribution

The paper presents the first average-case quasi-polynomial time algorithm for unimodular equivalence testing, along with new canonical forms and an acceleration strategy.

Findings

First average-case quasi-polynomial time algorithm for UP-equivalence

Polynomial-time complexity with extremely low failure probability

Proved equivalence of simplices via their pyramids

Abstract

Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UP-equivalence, which leads to the first average-case quasi-polynomial time algorithm, called \texttt{HEM}, for deciding the unimodular equivalence of $d$-dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than $2.5 \times 10^{-7}$. A key ingredient is the introduction of the \emph{permuted Hermite normal form} and its associated \emph{pattern group}, which streamlines the UP-equivalence test by comparing canonical forms derived from induced coset representatives. We also present an acceleration strategy based on Smith normal forms. As a theoretical by-product, we prove that two full-dimensional integral simplices are unimodularly equivalent if and only if their $n$-dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.