Lagrangian Duality in Online Scheduling with Resource Augmentation and Speed Scaling

We present an unified approach to study online scheduling problems in the resource augmentation/speed scaling models. Potential function method is extensively used for analyzing algorithms in these models; however, they yields little insight on how to construct potential functions and how to design algorithms for related problems. In the paper, we generalize and strengthen the dual-fitting technique proposed by Anand et al. [1]. The approach consists of considering a possibly non-convex relaxation and its Lagrangian dual; then constructing dual variables such that the Lagrangian dual has objective value within a desired factor of the primal optimum. The competitive ratio follows by the standard Lagrangian weak duality. This approach is simple yet powerful and it is seemingly a right tool to study problems with resource augmentation or speed scaling. We illustrate the approach through the following results. 1. We revisit algorithms EQUI and LAPS in Non-clairvoyant Scheduling to minimize total flow-time. We give simple analyses to prove known facts on the competitiveness of such algorithms. Not only are the analyses much simpler than the previous ones, they also explain why LAPS is a natural extension of EQUI to design a scalable algorithm for the problem. 2. We consider the online scheduling problem to minimize total weighted flow-time plus energy where the energy power f(s) is a function of speed s and is given by s for α ≥ 1. For a single machine, we showed an improved competitive ratio for a non-clairvoyant memoryless algorithm. For unrelated machines, we give an O(α/ logα)-competitive algorithm. The currently best algorithm for unrelated machines is O(α)-competitive. 3. We consider the online scheduling problem on unrelated machines with the objective of minimizing ∑ i,j wijf(Fj) where Fj is the flowtime of job j and f is an arbitrary non-decreasing cost function with some nice properties. We present an algorithm which is 1 1−3ǫ -speed,