Primal-Dual and Dual-Fitting Analysis of Online Scheduling Algorithms for Generalized Flow Time Problems

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 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.

Optimal Resource Allocation for Elastic and Inelastic Jobs

“…Using standard dualfitting techniques for worst-case analysis (e.g. [4,5]), we show SRPT-k is a 4-approximation for the objective of minimizing mean response time. This demonstrates the need for stochastic modeling and analysis.…”

“…Note, we leverage the fact µ I ≥ µ E in (5). If µ E > µ I , then µ I − µ E would be negative, so we would not be able establish a relationship like (5).…”

Competitive Algorithms from Competitive Equilibria

“…In the online setting, several works (Chadha et al 2009; give competitive clairvoyant algorithms for the weighted flow time objective on unrelated machines. Linear (or convex) programs and dual fitting approaches have been popular for online scheduling Gupta et al 2012b;Devanur and Huang 2014;Angelopoulos et al 2015); for an overview of online scheduling see Pruhs et al (2004). Though (Azar et al 2013) study a general online packing and covering framework, it does not capture temporal aspects of scheduling and is very different from our framework.…”

Optimal Speed Scaling with a Solar Cell

“…Job j is released at time 1 with deadline 2 and work 2. The energy optimal schedule runs job j at speed 2 during the time interval [1,2] and job j at speed 1 during the time intervals [0, 1] and [2,4]. It needs recharge rate R = 2.5.…”

“…Our algorithm can be viewed as a homotopic optimization algorithm that maintains an energy optimal schedule while the recharge rate is continuously decreased. Similar approaches for other speed scaling problems have been used in [1,2,6,7]. The resulting combinatorial algorithm exposes interesting structural properties and relations to be maintained while decreasing the recharge rate and adapting the schedule, not unlike (but much more complex than) the homotopic algorithm from [2].…”