LLL-reduction for Integer Knapsacks

TL;DR

This paper introduces a polynomial-time algorithm based on LLL reduction for solving integer knapsack problems under specific constraints on the target vector, improving computational efficiency.

Contribution

It presents a novel LLL-based method that efficiently solves integer knapsack problems with certain regularity assumptions on the matrix.

Findings

Algorithm operates in polynomial time

Effective for problems with constrained target vectors

Enhances existing integer programming techniques

Abstract

Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point exists. We obtain a LLL-based polynomial time algorithm that solves the problem subject to a constraint on the location of the vector b.