Solving Sparse Integer Linear Systems

TL;DR

This paper introduces a novel algorithm for efficiently solving sparse integer linear systems using p-adic lifting and structured block matrices, demonstrating practical improvements over existing methods.

Contribution

The paper presents a new sparse integer linear system solver combining p-adic lifting with structured block matrices, achieving sub-cubic complexity under certain conjectures.

Findings

Sub-cubic complexity achieved in theory

Practical implementation shows improved performance

Demonstrated benefits over previous methods

Abstract

We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.