Improved Bounds for Scheduling Flows under Endpoint Capacity Constraints

TL;DR

This paper advances the theoretical understanding of flow scheduling under node capacity constraints by providing improved bounds and algorithms for response time objectives, with and without resource augmentation.

Contribution

It introduces new algorithms that achieve better competitive ratios for maximum response time under various resource augmentation settings.

Findings

Proportional Allocation algorithm achieves (1/ε)-competitive ratio with (1+ε) resource augmentation.

Batch Decomposition algorithm is 2-competitive for maximum response time with resource augmentation 2.

Bounds for simultaneous approximation of average and maximum response times are established.

Abstract

We study flow scheduling under node capacity constraints. We are given capacitated nodes and an online sequence of jobs, each with a release time and a demand to be routed between two nodes. A schedule specifies which jobs are routed in each step, guaranteeing that the total demand on a node in any step is at most its capacity. A key metric in this scenario is response time: the time between a job's release and its completion. Prior work shows no un-augmented algorithm is competitive for average response time, and that a constant factor competitive ratio is achievable with augmentation exceeding 2 (Dinitz-Moseley Infocom 2020). For maximum response time, the best known result is a 2-competitive algorithm with a augmentation 4 (Jahanjou et al SPAA 2020). We improve these bounds under various response time objectives. We show that, without resource augmentation, the best competitive ratio for maximum response time is $\Omega(n)$, where $n$ is the number of nodes. Our Proportional Allocation algorithm uses $(1+\varepsilon)$ resource augmentation to achieve a $(1/\varepsilon)$-competitive ratio in the setting with general demands and capacities, and splittable jobs. Our Batch Decomposition algorithm is $2$-competitive (resp., optimal) for maximum response time using resource augmentation 2 (resp., 4) in the setting with unit demands and capacities, and unsplittable jobs. We also derive bounds for the simultaneous approximation of average and maximum response time metrics.