The complexity of geometric scaling

TL;DR

This paper investigates the complexity of geometric scaling methods for integer optimization over 0/1-polytopes, establishing tight bounds on the number of oracle calls needed in worst-case scenarios.

Contribution

It provides tight upper bounds on the number of augmentation oracle calls for geometric scaling, considering both maximum ratio and approximate variants.

Findings

Upper bounds are tight for worst-case scenarios.

Bounds depend on the dimension and the infinity norm of the vector c.

The analysis applies to both exact and approximate augmentation methods.

Abstract

Geometric scaling, introduced by Schulz and Weismantel in 2002, solves the integer optimization problem $\max \{c\mathord{\cdot}x: x \in P \cap \mathbb Z^n\}$ by means of primal augmentations, where $P \subset \mathbb R^n$ is a polytope. We restrict ourselves to the important case when $P$ is a $0/1$-polytope. Schulz and Weismantel showed that no more than $O(n \log n \|c\|_\infty)$ calls to an augmentation oracle are required. This upper bound can be improved to $O(n \log \|c\|_\infty)$ using the early-stopping policy proposed in 2018 by Le Bodic, Pavelka, Pfetsch, and Pokutta. Considering both the maximum ratio augmentation variant of the method as well as its approximate version, we show that these upper bounds are essentially tight by maximizing over a $n$-dimensional simplex with vectors $c$ such that $\|c\|_\infty$ is either $n$ or $2^n$.