Column basis reduction and decomposable knapsack problems

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b′≤Ax≤bx∈Znb′≤Ax≤bx∈Zn with b′≤(AU)y≤by∈Zn,b′≤(AU)y≤by∈Zn, where UU is a unimodular matrix computed via basis reduction  , to make the columns of AUAU short (i.e. have small Euclidean norm), and nearly orthogonal (see e.g. [Arjen K. Lenstra, Hendrik W. Lenstra, Jr., László Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen 261 (1982) 515–534; Ravi Kannan, Minkowski’s convex body theorem and integer programming, Mathematics of Operations Research 12 (3) (1987) 415–440]). 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 pM+rpM+r, with p>0p>0 and rr integral vectors, and MM 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 pxpx instead; and•that branching on the last few variables in either the rangespace or the AHL reformulations is equivalent to branching on pxpx 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.