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Scheduling with Machine Conflicts

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Approximation and Online Algorithms (WAOA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13538))

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Abstract

We study the scheduling problem of makespan minimization with machine conflicts that arise in various settings, e.g., shared resources for pre- and post-processing of tasks or spatial restrictions. In this context, each job has a blocking time before and after its processing time, i.e., three parameters. Given a set of jobs, a set of machines, and a graph representing machine conflicts, the problem SchedulingWithMachineConflicts (smc), asks for a conflict-free schedule of minimum makespan in which the blocking times of no two jobs intersect on conflicting machines.

We show that, unless \(\text {P} =\text {NP} \), smc on m machines does not allow for a \(\mathcal {O}(m^{1-\varepsilon })\)-approximation algorithm for any \(\varepsilon >0\), even in the case of identical jobs and every choice of fixed positive parameters, including the unit case. Complementary, we provide approximation algorithms when a suitable collection of independent sets is given. Finally, we present polynomial time algorithms to solve the problem for the case of unit jobs smc-Unit on special graph classes. As our main result, we solve smc-Unit for bipartite graphs by using structural insights for conflict graphs of star forests. As the set of active machines at each point in time induces a bipartite graph, the insights yield a local optimality criterion.

D. Schmidt genannt Waldschmidt—was funded by the DFG under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

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Author information

Authors and Affiliations

  1. School of Business and Economics, Maastricht University, Maastricht, The Netherlands

    Moritz Buchem

  2. Department of Computer Science, TU Braunschweig, Braunschweig, Germany

    Linda Kleist

  3. Institute for Mathematics, TU Berlin, Berlin, Germany

    Daniel Schmidt genannt Waldschmidt

Authors
  1. Moritz Buchem
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  2. Linda Kleist
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Correspondence to Daniel Schmidt genannt Waldschmidt .

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Editors and Affiliations

  1. Aalto University, Espoo, Finland

    Parinya Chalermsook

  2. Shanghai University of Finance and Economics, Shanghai, China

    Bundit Laekhanukit

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Buchem, M., Kleist, L., Schmidt genannt Waldschmidt, D. (2022). Scheduling with Machine Conflicts. In: Chalermsook, P., Laekhanukit, B. (eds) Approximation and Online Algorithms. WAOA 2022. Lecture Notes in Computer Science, vol 13538. Springer, Cham. https://doi.org/10.1007/978-3-031-18367-6_3

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  • DOI: https://doi.org/10.1007/978-3-031-18367-6_3

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