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Completing Partial Schedules for Open Shop with Unit Processing Times and Routing

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Computer Science – Theory and Applications (CSR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9691))

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Abstract

Open Shop is a classical scheduling problem: given a set \(\mathcal J\) of jobs and a set \(\mathcal M\) of machines, find a minimum-makespan schedule to process each job \(J_i\in \mathcal J\) on each machine \(M_q\in \mathcal M\) for a given amount \(p_{iq}\) of time such that each machine processes only one job at a time and each job is processed by only one machine at a time. In Routing Open Shop, the jobs are located in the vertices of an edge-weighted graph \(\mathcal G=(V,E)\) whose edge weights determine the time needed for the machines to travel between jobs. The travel times also have a natural interpretation as sequence-dependent family setup times. Routing Open Shop is NP-hard for \(|V|=|\mathcal M|=2\). For the special case with unit processing times \(p_{iq}=1\), we exploit Galvin’s theorem about list-coloring edges of bipartite graphs to prove a theorem that gives a sufficient condition for the completability of partial schedules. Exploiting this schedule completion theorem and integer linear programming, we show that Routing Open Shop with unit processing times is solvable in  time, that is, fixed-parameter tractable parameterized by \(|V|+|\mathcal M|\). Various upper bounds shown using the schedule completion theorem suggest it to be likewise beneficial for the development of approximation algorithms.

R. van Bevern—Supported by the Russian Foundation for Basic Research (RFBR) under research project 16-31-60007 mol_a_dk.

A.V. Pyatkin—Supported by the RFBR under research projects 15-01-00462 and 15-01-00976.

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Notes

  1. 1.

    We abstain from a more detailed running time analysis since no such analysis is available for the forthcoming application of Theorem 3.6 (yet).

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Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russian Federation

    René van Bevern & Artem V. Pyatkin

  2. Sobolev Institute of Mathematics, Novosibirsk, Russian Federation

    Artem V. Pyatkin

Authors
  1. René van Bevern
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Correspondence to René van Bevern .

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Editors and Affiliations

  1. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

  2. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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van Bevern, R., Pyatkin, A.V. (2016). Completing Partial Schedules for Open Shop with Unit Processing Times and Routing. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-34171-2_6

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