Approximation Algorithms for Scheduling under Non-Uniform Machine-Dependent Delays
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We consider the problem of scheduling precedence-constrained jobs on heterogenous machines in the presence of non-uniform machine-dependent communication delays. We are given as input a set of n unit size precedence-constrained jobs, and a set of m related machines each with size m_i (machine i can execute at most m_i jobs at any time). Each machine i also has an associated in-delay ρ^in_i and out-delay ρ^out_i. For any job v, machine i, and time t, we say that v is available to machine i at time t if v is completed on i before time t or on any machine j before time t - (ρ^in_i + ρ^out_j). If job v is scheduled at time t on machine i, then all of its predecessors must be available to i by time t. The objective is to construct a schedule that minimizes makespan, which is the maximum completion time over all jobs. We consider schedules which allow duplication of jobs as well as schedules which do not. When duplication is allowed, we provide an asymptotic polylog(n)-approximation algorithm; it is a true approximation if the makespan also accounts for the time to communicate the jobs to the machines and for the time to communicate the results out. For no-duplication schedules, we also obtain an asymptotic polylog(n)-approximation via a reduction to the case with duplication, and a true polylog(n)-approximation for symmetric delays (ρ^in_i = ρ^out_i for all machines i). These results represent the first polylogarithmic approximation algorithms for scheduling with non-uniform communication delays.
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