Deterministic Monotone Algorithms for Scheduling on Related Machines

We consider the problem of designing monotone deterministic algorithms for scheduling tasks on related machines in order to minimize the makespan. Several recent papers showed that monotonicity is a fundamental property to design truthful mechanisms for this scheduling problem. We give both theoretical and experimental results. For the case of two machines, when speeds of the machines are restricted to be powers of a given constant c > 0, we prove that algorithm Largest Processing Time is monotone for any c ≥ 2 while it is not monotone for c ≤ 1.78; algorithm List Scheduling, instead, is monotone only for c > 2. For the case of m machines we restrict our attention to the class of “greedy-like” monotone algorithms defined in [AP04]. We propose the greedy–like algorithm Uniform_RR and we prove that it is monotone when speeds are powers of a given integer constant c >0 and it obtains an approximation ratio that is not worse than algorithm Uniform, proposed in [AP04]. We also experimentally compare performances of Uniform, Uniform_RR, LPT, and several other monotone and greedy–like heuristics.

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