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

  • René van Bevern
  • Artem V. Pyatkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

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 Open image in new window  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.

Keywords

NP-hard scheduling problem Fixed-parameter algorithm Edge list-coloring Sequence-dependent family or batch setup times 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Sobolev Institute of MathematicsNovosibirskRussian Federation

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