The complexity of geometric scaling

Geometric scaling, introduced by Schulz and Weismantel in 2002, solves the integer optimization problem max{c.x : x is an element of P boolean AND Z(n)} by means of primal augmentations, where P subset of 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(2) n parallel to c parallel to(infinity)) calls to an augmentation oracle are required. This upper bound can be improved to O(n log(2) parallel to c parallel to(infinity)) 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 csuch that parallel to c parallel to(infinity) is either nor 2(n). (c) 2023 Elsevier B.V. All rights reserved.