Primal-dual and dual-fitting analysis of online scheduling algorithms for generalized flow-time problems

TL;DR

This paper introduces a duality-based analysis framework for online scheduling problems, improving competitive ratios and extending applicability to various generalized flow-time objectives without relying on potential functions.

Contribution

It develops a primal-dual analysis approach that bypasses potential functions, leading to improved competitive ratios for generalized flow-time scheduling problems.

Findings

Optimality of Highest-Density-First (HDF) for fractional weighted flow time

Extended analysis to concave and general cost functions

Improved competitive ratios for various scheduling settings

Abstract

We study online scheduling problems on a single processor that can be viewed as extensions of the well-studied problem of minimizing total weighted flow time. In particular, we provide a framework of analysis that is derived by duality properties, does not rely on potential functions and is applicable to a variety of scheduling problems. A key ingredient in our approach is bypassing the need for "black-box" rounding of fractional solutions, which yields improved competitive ratios. We begin with an interpretation of Highest-Density-First (HDF) as a primal-dual algorithm, and a corresponding proof that HDF is optimal for total fractional weighted flow time (and thus scalable for the integral objective). Building upon the salient ideas of the proof, we show how to apply and extend this analysis to the more general problem of minimizing $\sum_j w_j g(F_j)$, where $w_j$ is the job weight, $F_j$ is the flow time and $g$ is a non-decreasing cost function. Among other results, we present improved competitive ratios for the setting in which $g$ is a concave function, and the setting of same-density jobs but general cost functions. We further apply our framework of analysis to online weighted completion time with general cost functions as well as scheduling under polyhedral constraints.