Parallel Approximation and Integer Programming Reformulation

TL;DR

This paper demonstrates that in low-density knapsack problems, near-parallel integral vectors can be used to identify effective branching directions, leading to improved reformulation techniques with bounded integer width.

Contribution

The paper introduces a new approach using near-parallel integral vectors to analyze and improve reformulation techniques for low-density knapsack problems.

Findings

Bound on integer width along the last variable when density is low

Existence of a near-parallel integral vector as a good branching direction

Reformulations with reduced constraint matrices and bounded integer width

Abstract

We show that in a knapsack feasibility problem an integral vector $p$, which is short, and near parallel to the constraint vector gives a branching direction with small integer width. We use this result to analyze two computationally efficient reformulation techniques on low density knapsack problems. Both reformulations have a constraint matrix with columns reduced in the sense of Lenstra, Lenstra, and Lov\'asz. We prove an upper bound on the integer width along the last variable, which becomes 1, when the density is sufficiently small. In the proof we extract from the transformation matrices a vector which is near parallel to the constraint vector $a.$ The near parallel vector is a good branching direction in the original knapsack problem, and this transfers to the last variable in the reformulations.