Block-Structured Integer and Linear Programming in Strongly Polynomial and Near Linear Time

Cslovjecsek, Jana; Eisenbrand, Friedrich; Hunkenschröder, Christoph; Rohwedder, Lars; Weismantel, Robert

doi:10.1137/1.9781611976465.101

“…Then, in 2019, a near linear dependence in n was achieved by Jansen et al [21] as well as Eisenbrand et al [12]. Finally, in 2021, Cslovjecsek et al [10] further improved and parallelized the result. For an overview on recent results on n-fold IPs and related topics we refer to [12].…”

Section: Related Workmentioning

confidence: 97%

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Jansen

¹

,

Klein

²

,

Maack

³

et al. 2021

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Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

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“…Following the work of Eisenbrand and Weismantel [15], the Steinitz Lemma has been used in numerous projects; we point to [8,10,13,14,22,23] just to name a few. One area that benefits from the Steinitz Lemma is the study of integer programs with special sparse block-structures.…”

Section: Corollary 1 Let (U Imentioning

confidence: 99%

“…In our results, we assume t 0 , s 0 ≥ 1. 4-block integer programs generalize nfold integer programs, which occur when t 0 = 0, and 2-stage stochastic integer programs, which occur when s 0 = 0; see [10,18,20,24] and [23,26].…”

Section: Corollary 1 Let (U Imentioning

confidence: 99%

A Colorful Steinitz Lemma with Applications to Block Integer Programs

Oertel¹,

Paat²,

Weismantel³

2022

Preprint

Self Cite

0

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The Steinitz constant in dimension d is the smallest value c(d) such that for any norm on R d and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by c(d). Grinberg and Sevastyanov prove that c(d) ≤ d and that the bound of d is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences.Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.1 Introduction.Let • : R d → R be an arbitrary norm with corresponding unit ballGrinberg and Sevastyanov prove the following result on zero-sum sequences. We refer to

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“…A (4) A (5) A (6) A (7) A (8) A (2) A (3) A (1) Let s i be the number of columns that are used by matrices in the ith level of the tree (starting from level 0 at the leaves). Here we assume that the number of columns of matrices in the same level of the tree are all identical.…”

Section: Multi-stage Stochastic Ipsmentioning

confidence: 99%

“…This class of IPs has found numerous applications in the area of string algorithms [24], computational social choice [25] and scheduling [19,23]. State-ofthe-art algorithms compute a solution of an n-fold IP in time O(r 2 s+s 2 ) • (n, t) 1+o(1) [7,10,11,20,26], where is the largest entry in the matrices A and B.…”

Section: Introductionmentioning

confidence: 99%

About the complexity of two-stage stochastic IPs

Klein

¹

2021

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We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix $${\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}$$ A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix $$A \in {\mathbb {Z}}^{r \times s}$$ A ∈ Z r × s on the vertical line and n repetitions of a matrix $$B \in {\mathbb {Z}}^{r \times t}$$ B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $$\Delta $$ Δ , where $$\Delta $$ Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $$f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)$$ f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

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Block-Structured Integer and Linear Programming in Strongly Polynomial and Near Linear Time

Cited by 29 publications

References 0 publications

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Empowering the configuration-IP: new PTAS results for scheduling with setup times

A Colorful Steinitz Lemma with Applications to Block Integer Programs

About the complexity of two-stage stochastic IPs

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