Column basis reduction and decomposable knapsack problems

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem

b' ≤ Ax ≤ b, x ∈ Zⁿ

with

b' ≤ (AU)y ≤ b, y ∈ Zⁿ,

where U is a unimodular matrix computed via basis reduction, to make the columns of AU short (i.e., have small Euclidean norm), and nearly orthogonal. Our approach is termed column basis reduction, and the reformulation is called rangespace reformulation. It is motivated by the technique proposed for equality constrained IPs by Aardal, Hurkens, and Lenstra. We also propose a simplified method to compute their reformulation. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, Chvátal and Todd, Avis, Aardal and Lenstra, and Cornuéjols et al. They are knapsack problems with a constraint vector of the form a = pM + r, with p > 0 and r integral vectors, and M a large integer. If the parameters are suitably chosen in DKPs, we prove

hardness results, when branch-and-bound branching on individual variables is applied;
that they are easy, if one branches on the constraint px instead; and
that branching on the last few variables in either the rangespace or the AHL reformulations is equivalent to branching on px in the original problem.

We also provide recipes to generate such instances. Our computational study confirms that the behavior of the studied instances in practice is as predicted by the theory.

Column basis reduction and decomposable knapsack problems

Citation

Article