Lagrangian Duality based Algorithms in Online Scheduling

TL;DR

This paper introduces Lagrangian duality based algorithms for online energy-efficient scheduling, providing a unified primal-dual framework and dual-fitting analysis for various problem settings.

Contribution

It develops a novel primal-dual framework and dual-fitting approach for designing and analyzing online scheduling algorithms across diverse environments.

Findings

Duality-based algorithms achieve competitive ratios in online scheduling.

Primal-dual approach offers intuitive decision structures.

Dual-fitting analysis applies to non-convex relaxations.

Abstract

We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to derive dual programs for problems which could be formulated as convex assignment problems. The duals have intuitive structures as the ones in linear programming. The constraints of the duals explicitly indicate the online decisions and naturally lead to competitive algorithms. Second, we use a dual-fitting approach, which also based on the weak duality, to study problems which are unlikely to admit convex relaxations. Through the analysis, we show an interesting feature in which primal-dual gives idea for designing algorithms while the analysis is done by dual-fitting. We illustrate the advantages and the flexibility of the approaches through problems in different setting: from single machine to unrelated machine environments, from typical competitive analysis to the one with resource augmentation, from convex relaxations to non-convex relaxations.