FPT-algorithms for The Shortest Lattice Vector and Integer Linear Programming Problems

TL;DR

This paper introduces fixed-parameter tractable algorithms for specific cases of the shortest lattice vector and integer linear programming problems, focusing on matrices with certain rank properties, advancing computational methods in these areas.

Contribution

It develops FPT-algorithms for SVP and ILP when matrices are near square or lack singular rank sub-matrices, based on the maximal absolute value of rank minors.

Findings

FPT-algorithms for near-square matrices in SVP and ILP

Algorithms for matrices without singular rank sub-matrices

Enhanced computational approaches for special matrix cases

Abstract

In this paper, we present FPT-algorithms for special cases of the shortest vector problem (SVP) and the integer linear programming problem (ILP), when matrices included to the problems' formulations are near square. The main parameter is the maximal absolute value of rank minors of matrices included to the problem formulation. Additionally, we present FPT-algorithms with respect to the same main parameter for the problems, when the matrices have no singular rank sub-matrices.