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An ILP representation of a DAG scheduling problem

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Abstract

The use of integer linear programs for solving complex real-time scheduling problems is investigated. The problem of scheduling a workload represented as a directed acyclic graph (DAG) upon a dedicated multiprocessor platform is considered, in which each individual vertex of the DAG is assigned to a specific processor and the entire DAG is required to complete execution within a specified duration. A representation of this scheduling problem as a zero-one integer linear program is obtained. Some guidelines are proposed for identifying scheduling problems amenable to solution by converting to integer linear programs.

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Notes

  1. We point out that RASP is not only a decision problem in the sense that for instances determined to be schedulable, we seek to additionally determine a correct schedule. We will see that our algorithm for solving the decision version of RASP is constructive in that it shows schedulability by actually generating a schedule.

  2. See (Cormen et al. 2009, p. 1063) for a rigorous textbook description of such verification algorithms.

  3. We reiterate that the processors are assumed to be preemptive: it is not required that the job \(v_i\) execute throughout the duration \([s_i, f_i]\). Rather, \(s_i\) denotes the first instant at which the job begins to execute, and \(f_i\), the last instant at which it does so.

  4. We emphasize that these jobs may be considered as being independent: precedence constraints between them are separately represented by constraints of the form given in Expression 3 above.

  5. Note that although Expression 5 appears to already be in linear-constraint form, it is not quite so since it is only defined for some values of i and j—those for which \(s_i\le f_j\). But since the \(s_i\) and the \(f_j\) values are variables, we cannot a priori determine for which values of i and j constraints of the form Expression 5 need to be written—this will be determined by the values that get assigned to the \(s_i\)’s and the \(f_j\)’s in any solution to the ILP that we are constructing.

  6. A subtle error in this reasoning was pointed out by Ben-Amor (2021, Appendix A): while we desire that \(x_{ij}=1\) if \(s_i\le s_j\) and \(y_{ij}=1\) if \(f_i\le f_j\), these inequalities allow for the possibility that \(x_{ij}=0\) (\(y_{ij}=0\), respectively) when \(s_i=s_j\) (\(f_i=f_j\), resp.). Ben-Amor describes a simple and elegant fix using the absolute value function that is available on standard ILP solvers.

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Funding

Funded was provided by National Science Foundation Grants CNS-1814739 and CPS-1932530.

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  1. Washington University in St. Louis, CB 1045, St Louis, MO, 63130, USA

    Sanjoy Baruah

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  1. Sanjoy Baruah
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Correspondence to Sanjoy Baruah.

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Baruah, S. An ILP representation of a DAG scheduling problem. Real-Time Syst 58, 85–102 (2022). https://doi.org/10.1007/s11241-021-09370-7

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