Lagrangian Duality in Online Scheduling with Resource Augmentation and Speed Scaling

Abstract

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 ^α 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 ²)-competitive.

• 3 We consider the online scheduling problem on unrelated machines with the objective of minimizing ∑  _i,j w _ij f(F _j ) where F _j is the flow-time of job j and f is an arbitrary non-decreasing cost function with some nice properties. We present an algorithm which is $\frac{1}{1-3\epsilon}$-speed, $\frac{2K(\epsilon)}{\epsilon}$-competitive where K(ε) is a function depending on f and ε. The algorithm does not need to know the speed (1 + ε) a priori. A corollary is a (1 + ε)-speed, $\frac{k}{\epsilon^{1+1/k}}$-competitive algorithm (which does not know ε a priori) for the objective of minimizing the weighted ℓ _k -norm of flow-time.