Packing-based branch-and-bound for discrete malleable task scheduling

This paper addresses the problem of scheduling chain-like structures of tasks on a single multiprocessor resource. In fact, sub-tasks of unit-time length and predefined size are aggregated to composite tasks that have to be scheduled without preemption, but subject to flexibility concerning resource allocation. This setting most closely resembles the problem of malleable task scheduling, with sub-tasks being the smallest atomic unit of allocation. The specific type of malleability is realized using precedence constraints with minimum and maximum time lags. A bin packing model is established for this scheduling problem and a corresponding, dedicated branch-and-bound algorithm is devised, alongside problem-specific bound tightening, symmetry breaking and dominance concepts. The efficacy of the solution approach is demonstrated based on extensive computational experiments, including randomized instances, adapted benchmark instances from the literature, and a small real-world data set. In comparison to mixed-integer and constraint programming formulations, the new method is able to achieve a considerably higher percentage of optimal solutions at computation times that are up to orders of magnitude smaller.