LLL-reduction for integer knapsacks

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