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Unimodular Equivalence of Integral Simplices

Feihu Liu, Sihao Tao, Guoce Xin

TL;DR

This work addresses the problem of detecting unimodular equivalence between full-dimensional integral simplices by reducing it to unimodular permutation (UP) equivalence of vertex matrices. It introduces the HEM algorithm, a Hermite-normal-form–based approach that uses permuted Hermite normal forms and pattern groups to obtain canonical representatives, achieving average-case quasi-polynomial time and provable polynomial-time performance with failure probability below $2.5\times 10^{-7}$. A dimension-free resolution to a key open problem is provided: two $d$-dimensional simplices are unimodularly equivalent if and only if their $n$-dimensional pyramids are unimodularly equivalent, for all $d$ and $n\ge d$. The results are complemented by an average-case analysis, SNF-based acceleration, and practical considerations that link to Ehrhart theory and related matrix invariants, with potential implications for broader equivalence testing problems including graph isomorphism.

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.

Unimodular Equivalence of Integral Simplices

TL;DR

This work addresses the problem of detecting unimodular equivalence between full-dimensional integral simplices by reducing it to unimodular permutation (UP) equivalence of vertex matrices. It introduces the HEM algorithm, a Hermite-normal-form–based approach that uses permuted Hermite normal forms and pattern groups to obtain canonical representatives, achieving average-case quasi-polynomial time and provable polynomial-time performance with failure probability below . A dimension-free resolution to a key open problem is provided: two -dimensional simplices are unimodularly equivalent if and only if their -dimensional pyramids are unimodularly equivalent, for all and . The results are complemented by an average-case analysis, SNF-based acceleration, and practical considerations that link to Ehrhart theory and related matrix invariants, with potential implications for broader equivalence testing problems including graph isomorphism.

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 -dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than . 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 -dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.
Paper Structure (17 sections, 29 theorems, 99 equations, 1 table, 5 algorithms)

This paper contains 17 sections, 29 theorems, 99 equations, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Two major contributions
  4. Equivalence relations on matrices and simplices
  5. Equivalence relations on integer matrices
  6. Unimodular equivalence of integral simplices
  7. Unimodular equivalence checking algorithm
  8. Existing algorithms
  9. Definitions and implication chain.
  10. An algorithm via Hermite normal form
  11. Permuted Hermite normal form
  12. The HEM Algorithm
  13. Evaluation of practical utility
  14. Analysis of HEM
  15. Average-case complexity
  16. ...and 2 more sections

Key Result

Theorem 1.6

Let $\mathcal{P} = \mathrm{conv}(A)$ and $\mathcal{Q} = \mathrm{conv}(B)$ be full-dimensional integral simplices in $\mathbb{R}^d$ with identical lattice volume $v$, where $A, B \in \mathbb{Z}^{d \times (d+1)}$ are integer matrices whose columns represent the vertices. Then HEM decides the unimodula

Theorems & Definitions (62)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Definition 2.5
  • ...and 52 more