About the Kannan-Bachem algorithm

TL;DR

This paper analyzes the Kannan-Bachem algorithm for integer matrix reduction, identifies an obstacle in non-maximal rank cases, and proposes a simplified, efficient, and more general version of the algorithm.

Contribution

It provides a detailed analysis of the KB algorithm's obstacle in certain cases and introduces a simplified, more versatile version of the algorithm.

Findings

Identified an obstacle in the KB algorithm for non-maximal rank matrices.

Proposed a simple solution to overcome the obstacle.

Developed a more efficient and general organization of the KB algorithm.

Abstract

The Smith reduction is a basic tool when analyzing integer matrices up to equivalence, and the Kannan-Bachem (KB) algorithm is the first polynomial algorithm computing such a reduction. Using this algorithm in complicated situations where the rank of the studied matrix is not maximal revealed an unexpected obstacle in the algorithm. This difficulty is described, analyzed, a simple solution is given to overcome it, finally leading to a general organization of the KB algorithm, simpler than the original one, efficient and having a general scope.