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Online interval scheduling with a bounded number of failures

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Abstract

We consider the problem of scheduling intervals on m identical machines where each interval can be seen as a job with fixed start and end time. The goal is to accept a maximum cardinality subset of the given intervals and assign these intervals to the machines subject to the constraint that no two intervals assigned to the same machine overlap. We analyze an online version of this problem where, initially, a set of n potential intervals and an upper bound k on the number of failing intervals is given. If an interval fails, it can be accepted neither by the online algorithm nor by the adversary. An online algorithm learns that an interval fails at the time when it is supposed to be started. If a non-failing interval is accepted, it cannot be aborted and must be processed non-preemptively until completion. For different settings of this problem, we present deterministic and randomized online algorithms and prove lower bounds on the competitive ratio.

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Notes

  1. Naturally, the problem can be modeled as a longest path problem. Since the graph is directed and acyclic, this is equivalent to a shortest path problem (by reversing the signs of all arc weights).

  2. Formally, the number of machines m is also part of the instance. However, we will sometimes slightly abuse notation and say that an instance \(\sigma \) is processed on m machines.

  3. Note that it can happen that some of the intervals we ignored in the first iteration are not ignored anymore in the second iteration.

  4. Here, \(\hat{\sigma } \subseteq \sigma \) means that \(I(\hat{\sigma }) \subseteq I(\sigma )\) and \(F(\hat{\sigma }) \subseteq F(\sigma ) \cap I(\hat{\sigma })\).

  5. Forbidding an algorithm to accept some intervals does not change the argumentation presented in the paragraph subsequent to (9).

  6. The argumentation for Observation 2 carries over to \(\overline{\textsc {alg}^m}\).

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Acknowledgements

We would like to thank Tjark Vredeveld for his helpful comments on this work, especially regarding the deterministic algorithm for multiple machines.

The first author was partially supported by the German Research Foundation (DFG), Grant GRK 1703/1 “Resource Efficiency in Interorganizational Networks.”

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Authors and Affiliations

  1. Institute for Applied Stochastics and Operations Research, Clausthal University of Technology, Clausthal-Zellerfeld, Germany

    Marco Bender & Stephan Westphal

  2. Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany

    Clemens Thielen

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  1. Marco Bender
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  2. Clemens Thielen
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  3. Stephan Westphal
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Correspondence to Marco Bender.

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Bender, M., Thielen, C. & Westphal, S. Online interval scheduling with a bounded number of failures. J Sched 20, 443–457 (2017). https://doi.org/10.1007/s10951-016-0506-9

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