Abstract
We consider the problem of solving integer programs of the form min {c^⊺ x : Ax = b, x ∈ ℤ_{⩾ 0}}, where A is a multistage stochastic matrix in the following sense: the primal treedepth of A is bounded by a parameter d, which means that the columns of A can be organized into a rooted forest of depth at most d so that columns not bound by the ancestor/descendant relation do not have nonzero entries in the same row. We give an algorithm that solves this problem in fixedparameter time f(d,‖A‖_{∞})⋅ nlog^{𝒪(2^d)} n, where f is a computable function and n is the number of rows of A. The algorithm works in the strong model, where the running time only measures unit arithmetic operations on the input numbers and does not depend on their bitlength. This is the first fpt algorithm for multistage stochastic integer programming to achieve almost linear running time in the strong sense. For twostage stochastic integer programs, our algorithm works in time 2^{((r+s)‖A‖_∞)^{𝒪(r(r+s))}}⋅ nlog^{𝒪(rs)} n, which improves over previous methods both in terms of the polynomial factor and in terms of the dependence on r and s. In fact, for r = 1 the dependence on s is asymptotically almost tight assuming the Exponential Time Hypothesis. Our algorithm can be also parallelized: we give an implementation in the PRAM model that achieves running time f(d,‖A‖_{∞})⋅ log^{𝒪(2^d)} n using n processors.
The main conceptual ingredient in our algorithms is a new proximity result for multistage stochastic integer programs. We prove that if we consider an integer program P, say with a constraint matrix A, then for every optimum solution to the linear relaxation of P there exists an optimum (integral) solution to P that lies, in the 𝓁_{∞}norm, within distance bounded by a function of ‖A‖_{∞} and the primal treedepth of A. On the way to achieve this result, we prove a generalization and considerable improvement of a structural result of Klein for multistage stochastic integer programs. Once the proximity results are established, this allows us to apply a treedepthbased branching strategy guided by an optimum solution to the linear relaxation.
BibTeX  Entry
@InProceedings{cslovjecsek_et_al:LIPIcs.ESA.2021.33,
author = {Cslovjecsek, Jana and Eisenbrand, Friedrich and Pilipczuk, Micha{\l} and Venzin, Moritz and Weismantel, Robert},
title = {{Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {33:133:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772044},
ISSN = {18688969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14614},
URN = {urn:nbn:de:0030drops146146},
doi = {10.4230/LIPIcs.ESA.2021.33},
annote = {Keywords: parameterized algorithm, multistage stochastic programming, proximity}
}
Keywords: 

parameterized algorithm, multistage stochastic programming, proximity 
Collection: 

29th Annual European Symposium on Algorithms (ESA 2021) 
Issue Date: 

2021 
Date of publication: 

31.08.2021 