Efficient Algorithms for Scheduling Moldable Tasks

We study the problem of scheduling n independent moldable tasks on m processors that arises in large-scale parallel computations. When tasks are monotonic, the best known result is a (3/2+ϵ)-approximation algorithm for makespan minimization with a complexity linear in n and polynomial in logm and 1/ϵ where ϵ is arbitrarily small. We propose a new perspective of the existing speedup models: the speedup of a task T_j is linear when the number p of assigned processors is small (up to a threshold δ_j) while it presents monotonicity when p ranges in [δ_j, k_j]; the bound k_j indicates an unacceptable overhead when parallelizing on too many processors. The generality of this model is proved to be between the classic monotonic and linear-speedup models. For any given integer δ≥ 5, let u=⌈√(δ)⌉-1≥ 2. In this paper, we propose a 1/θ(δ) (1+ϵ)-approximation algorithm for makespan minimization where θ(δ) = u+1/u+2( 1- k/m) (m≫ k). As a by-product, we also propose a θ(δ)-approximation algorithm for throughput maximization with a common deadline.