Scheduling jobs with communication delays: Using infeasible solutions for approximation

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In the last few years, multi-processor scheduling with interprocessor communication delays has received increasing attention. This is due to the more realistic constraints in modeling parallel processor systems.

Most research in this vein is concerned with the makespan criterion. We contribute to this work by presenting a new and simple (2−1/m)-approximation algorithm for scheduling to minimize the makespan on identical parallel processors subject to series-parallel precedence constraints and both unit processing times and communication delays. This meets the best known performance guarantee for the same problem but without communication delays. For the same problem but with (non-trivial) release dates, arbitrary precedence constraints, arbitrary processing times and “locally small” communication delays we obtain a simple 7/3-approximation algorithm compared with the involved (7/3−4/3m)-approximation algorithm by Hanen and Munier for the case with identical release dates.

Another quite important goal in real-world scheduling is to optimize average performance. Very recently, there have been significant developments in computing nearly optimal schedules for several classic processor scheduling models to minimize the average weighted completion time. In this paper, we study for the first time scheduling with communication delays to minimize the average weighted completion time. Specifically, based on an LP relaxation we give the first constant-factor polynomial-time approximation algorithm for scheduling identical parallel processors subject to release dates and locally small communication delays. Moreover, the optimal LP value provides a lower bound on the optimum with the same worst-case performance guarantee.

The common underlying idea of our algorithms is to compute first a schedule that regards all constraints except for the processor restrictions. This schedule is then used to construct a provable good feasible schedule for a given number of processors and as a tool in the analysis of our algorithms. Complementing our approximation results, we also show that minimizing the makespan on an unrestricted number of identical parallel processors subject to series-parallel precedence constraints, unit-time jobs, and zero-one communication delays is NP-hard.