Polynomially Solvable Instances of the Shortest and Closest Vector Problems with Applications to Compute-and-Forward

TL;DR

This paper identifies a specific class of instances for the SVP related to Compute-and-Forward that can be solved efficiently, extending to approximation algorithms for related lattice problems.

Contribution

It introduces a polynomial-time algorithm for a particular SVP instance in Compute-and-Forward and extends results to Integer-Forcing and broader lattice classes.

Findings

Efficient polynomial-time solution for a specific SVP instance.

Extension of results to Integer-Forcing.

Approximation algorithms for SVP and CVP in broader lattice classes.

Abstract

A particular instance of the Shortest Vector Problem (SVP) appears in the context of Compute-and-Forward. Despite the NP-hardness of the SVP, we will show that this certain instance can be solved in complexity order $O(n\psi\log(n\psi))$ where $\psi = \sqrt{P\|{\bf h}\|^2+1}$ depends on the transmission power and the norm of the channel vector. We will then extend our results to Integer-Forcing and finally, introduce a more general class of lattices for which the SVP and the and the Closest Vector Problem (CVP) can be approximated within a constant factor.