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

Spyros Angelopoulos; Giorgio Lucarelli; Thang Nguyen Kim

doi:10.1007/s00453-019-00583-8

Algorithmica

September 2019, Volume 81, Issue 9, pp 3391–3421 | Cite as

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

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Spyros Angelopoulos
Giorgio Lucarelli
Nguyen Kim Thang

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First Online: 11 May 2019

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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.

Keywords

Online algorithms Primal–dual Dual-fitting Generalized flow-time Scheduling

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Notes

Acknowledgements

We would like to thank an anonymous referee for pointing out an error in an earlier version of this paper. Spyros Angelopoulos is supported by project ANR-11-BS02-0015 “New Techniques in Online Computation (NeTOC)”. Giorgio Lucarelli is supported by the ANR project Moebus (Grant No. ANR-13-INFR-0001). Nguyen Kim Thang supported by the FMJH program Gaspard Monge in optimization and operations research and by EDF.

LP-Formulation of Fractional JDGFP

We argue that the following LP is a relaxation of JDGFP.

$$\begin{aligned} \min \sum _{j \in \mathcal {J}} \delta _j \int _{r_j}^{\infty } g_j(t-r_j)&x_j(t) dt \\ \int _{r_j}^{\infty } x_j(t) dt&\ge p_j&\quad \forall j \in \mathcal {J} \\ \sum _{j \in \mathcal {J}} x_j(t)&\le 1&\quad \forall t \ge 0 \\ x_j(t)&\ge 0&\qquad \forall j \in \mathcal {J}, t \ge 0 \end{aligned}$$

Consider a job $j \in \mathcal {J}$. By definition, the fractional weighted cost of a job $j \in \mathcal {J}$ is equal to

$$\begin{aligned} \int _{r_j}^{\infty } w_j(t) g_j'(t-r_j) dt= & {} \frac{w_j}{p_j} \int _{r_j}^{\infty } q_j(t) g_j'(t-r_j) dt\\= & {} \frac{w_j}{p_j} \int _{r_j}^{\infty } \left( g_j'(t-r_j) \int _{u=t}^{\infty } x_j(u) du \right) dt \end{aligned}$$

By changing the order of the integrals we get

$$\begin{aligned} \int _{r_j}^{\infty } w_j(t) g_j'(t-r_j) dt= & {} \frac{w_j}{p_j} \int _{u=r_j}^{\infty } \left( x_j(u) \int _{t=r_j}^{u} g_j'(t-r_j) dt\right) du\\= & {} \frac{w_j}{p_j} \int _{u=r_j}^{\infty } g_j(u-r_j) x_j(u) du \end{aligned}$$

$\square $

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Authors and Affiliations

Spyros Angelopoulos
- 1
Giorgio Lucarelli
- 2
- 3
Email author
Nguyen Kim Thang
- 4

1.LIP6Sorbonne Université, CNRS, Laboratoire d’informatique de Paris 6ParisFrance
2.LIGUniversity Grenoble-AlpesGrenobleFrance
3.LCOMSUniversité de LorraineMetzFrance
4.IBISCUniv Evry, Université Paris-SaclayEvryFrance

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

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