Abstract
Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4block nfold integer programs, which are the most general class considered so far. A 4block nfold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the socalled Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the πβ or π_βnorm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4block nfold IP has Graver elements of π_βnorm at most πͺ_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and πͺ_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to πͺ_FPT(1), perhaps at least in some restricted cases.
We prove that the π_βnorm of the Graver elements of 4block nfold IP is upper bounded by πͺ_FPT(n^{s_D}), improving significantly over the previous bound πͺ_FPT(n^{2^{s_D}}). We also provide a matching lower bound of Ξ©(n^{s_D}) which even holds for arbitrary nonzero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4block nfold in which C is a zero matrix, called 3block nfold IP. We show that while the π_βnorm of its Graver elements is Ξ©(n^{s_D}), there exists a different decomposition into lattice elements whose π_βnorm is bounded by πͺ_FPT(1), which allows us to provide improved upper bounds on the π_βnorm of Graver elements for 3block nfold IP. The key difference between the respective decompositions is that a Graver basis guarantees a signcompatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3block nfold IP and 4block IP, and our lower bounds strongly hint at parameterized hardness of 4block and even 3block nfold IP. Furthermore, we show that 3block nfold IP is without loss of generality in the sense that 4block nfold IP can be solved in FPT oracle time by taking an algorithm for 3block nfold IP as an oracle.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs:2020:12899,
author = {Lin Chen and Martin Kouteck{\'y} and Lei Xu and Weidong Shi},
title = {{New Bounds on Augmenting Steps of BlockStructured Integer Programs}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {33:133:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771627},
ISSN = {18688969},
year = {2020},
volume = {173},
editor = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12899},
URN = {urn:nbn:de:0030drops128994},
doi = {10.4230/LIPIcs.ESA.2020.33},
annote = {Keywords: Integer Programming, Graver basis, Fixed parameter tractable}
}
Keywords: 

Integer Programming, Graver basis, Fixed parameter tractable 
Collection: 

28th Annual European Symposium on Algorithms (ESA 2020) 
Issue Date: 

2020 
Date of publication: 

26.08.2020 