Column basis reduction, and decomposable knapsack problems

TL;DR

This paper introduces a simple preconditioning method for integer programming called rangespace reformulation, and studies a class of decomposable knapsack problems, showing how different branching strategies affect their complexity.

Contribution

It proposes a new rangespace reformulation technique using basis reduction and analyzes the complexity of decomposable knapsack problems under various branching strategies.

Findings

Rangespace reformulation makes columns short and nearly orthogonal.

Branching on the constraint px simplifies certain knapsack problems.

Computational results confirm theoretical predictions about problem difficulty.

Abstract

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \in Z^n with b' <= AUy <= b, y \in Z^n, where U is a unimodular matrix computed via basis reduction, to make the columns of $AU$ short and nearly orthogonal. The reformulation is called rangespace reformulation. It is motivated by the reformulation technique proposed for equality constrained IPs by Aardal, Hurkens and Lenstra. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, Chvatal and Todd, Avis, Aardal and Lenstra, and Cornuejols et al. DKPs are knapsack problems with a constraint vector of the form $pM + r, $ with $p >0$ and $r$ integral vectors, and $M$ a large integer. If the parameters are suitably chosen in DKPs, we prove 1) hardness results for these problems, when branch-and-bound branching on individual variables is applied; 2) that they are easy, if one branches on the constraint $px$ instead; and 3) 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 theoretical results.