Online Primal-Dual For Non-linear Optimization with Applications to Speed Scaling

TL;DR

This paper develops an online primal-dual framework for nonlinear load-balancing problems, including speed scaling, providing new algorithms and simpler proofs for complex scheduling and routing problems.

Contribution

It introduces a novel online primal-dual approach to nonlinear load balancing, connecting speed scaling with load balancing problems, and offers new algorithms and analysis techniques.

Findings

Unified primal-dual analysis for nonlinear load balancing

New algorithms for speed scaling problems

Simpler proofs for existing results in speed scaling

Abstract

We reinterpret some online greedy algorithms for a class of nonlinear "load-balancing" problems as solving a mathematical program online. For example, we consider the problem of assigning jobs to (unrelated) machines to minimize the sum of the alpha^{th}-powers of the loads plus assignment costs (the online Generalized Assignment Problem); or choosing paths to connect terminal pairs to minimize the alpha^{th}-powers of the edge loads (online routing with speed-scalable routers). We give analyses of these online algorithms using the dual of the primal program as a lower bound for the optimal algorithm, much in the spirit of online primal-dual results for linear problems. We then observe that a wide class of uni-processor speed scaling problems (with essentially arbitrary scheduling objectives) can be viewed as such load balancing problems with linear assignment costs. This connection gives new algorithms for problems that had resisted solutions using the dominant potential function approaches used in the speed scaling literature, as well as alternate, cleaner proofs for other known results.