Staircase compatibility and its applications in scheduling and piecewise linearization

Bärmann, Andreas; Gellermann, Thorsten; Merkert, Maximilian; Schneider, Oskar

doi:10.1016/j.disopt.2018.04.001

Cited by 11 publications

(10 citation statements)

References 9 publications

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“…As the authors of [BGMS18] have shown, the CPMC polytope can be represented via a totally unimodular system of polynomial size if the considered instance of CPMC has the staircase property. In the following, we will give an alternative, shorter proof for their convex-hull result by using Zuckerberg's method.…”

Section: Discussionmentioning

confidence: 99%

“…, V m } be the corresponding partition of the node set V. The clique problem with multiple-choice constraints (CMPC) asks to find a clique of cardinality m in G. While it is NP-complete in general to decide if such a clique exists (see [BGM20]), there are several relevant special cases where this is possible in polynomial time. These include CPMC under staircase compatibility ( [BGMS18]) and CPMC under a cycle-free dependency graph ( [BGM20]). The referenced works give complete convex-hull descriptions for these two cases.…”

Section: Zuckerberg Proofs Via Feasibility Subproblemsmentioning

confidence: 99%

“…The CPMC polytope is the convex hull of all incidence vectors of m-cliques in G. In the online supplement [BS20], we will reprove the result from [BGMS18] that staircase compatibility allows for totally unimodular formulations of polynomial size for the CPMC polytope.…”

Section: Zuckerberg Proofs Via Feasibility Subproblemsmentioning

confidence: 99%

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Set characterizations and convex extensions for geometric convex-hull proofs

Bärmann,

Schneider

2021

Preprint

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In the present work, we consider Zuckerberg's method for geometric convex-hull proofs introduced in [Geometric proofs for convex hull defining formulations, Operations Research Letters 44(5), 625-629 (2016)]. It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg's proof technique, building on ideas from [Extended formulations for convex hulls of some bilinear functions, Discrete Optimization 36, 100569 (2020)]. We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg's proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

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Section: Discussionmentioning

confidence: 99%

Section: Zuckerberg Proofs Via Feasibility Subproblemsmentioning

confidence: 99%

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Set characterizations and convex extensions for geometric convex-hull proofs

Bärmann,

Schneider

2021

Preprint

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show abstract

“…The clique problem with multiplechoice constraints (CMPC) asks to find a clique of cardinality m in G. While it is NP-complete in general to decide if such a clique exists (see [5]), there are several relevant special cases where this is possible in polynomial time. These include CPMC under staircase compatibility [3,4] and CPMC under a cycle-free dependency graph [5]. The referenced works give complete convex-hull descriptions for these two cases.…”

Section: Zuckerberg Proofs Via Feasibility Subproblemsmentioning

confidence: 99%

“…The CPMC polytope is the convex hull of all incidence vectors of m-cliques in G. In the online supplement [7], we will reprove the result from [3] that staircase compatibility allows for totally unimodular formulations of polynomial size for the CPMC polytope. Here we consider the case where there are no cyclic dependencies between the subsets V i .…”

Section: Zuckerberg Proofs Via Feasibility Subproblemsmentioning

confidence: 99%

Set characterizations and convex extensions for geometric convex-hull proofs

Bärmann

Schneider

2021

Math. Program.

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In the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 44(5):625–629, 2016). It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg’s proof technique, building on ideas from Gupte et al. (Discrete Optim 36:100569, 2020). We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg’s proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

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EETTlib—Energy‐efficient train timetabling library

et al. 2022

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We introduce EETTlib, an instance library for the Energy‐Efficient Train Timetabling problem. The task in this problem is to adjust a given timetable draft such that the energy consumption of the resulting railway traffic is minimized. To this end, the departure times of the trains can be slightly, and their velocity profiles on each trip can be modified. We provide real‐world data originating from two research projects in this field, one with Deutsche Bahn AG, the most important railway company in Germany, the other with VAG Verkehrs‐Aktiengesellschaft, the operator of public transport in the city of Nürnberg, Germany. In both cases, our library contains representative data on the relevant operational constraints and supports various possible choices for the objective function with respect to energy‐efficiency. The resulting benchmark instances can be used by the scheduling and timetabling community to improve their models and algorithms. They are available under https://www.eettlib.fau.de.

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Staircase compatibility and its applications in scheduling and piecewise linearization

Cited by 11 publications

References 9 publications

Set characterizations and convex extensions for geometric convex-hull proofs

Set characterizations and convex extensions for geometric convex-hull proofs

Set characterizations and convex extensions for geometric convex-hull proofs

EETTlib—Energy‐efficient train timetabling library

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