Improved Algorithms for Scheduling Unsplittable Flows on Paths

TL;DR

This paper develops improved offline and online algorithms for scheduling unsplittable flows on paths, especially without the no bottleneck assumption, achieving better approximation and competitiveness bounds.

Contribution

It introduces new algorithms for rufpp without the NBA, including an $O( ext{log}(1/(1- ext{alpha})))$-approximation offline algorithm and an $O( ext{log log} c_{max})$-competitive online algorithm, improving previous bounds.

Findings

Online algorithm is $O( ext{log log} c_{max})$-competitive.

Offline approximation is $O( ext{log}(1/(1- ext{alpha})))$ for $ ext{alpha}$-small instances.

Achieves $O( ext{min}( ext{log} n, ext{log} m, ext{log log} c_{max}))$ approximation.

Abstract

In this paper, we investigate offline and online algorithms for rufpp, the problem of minimizing the number of rounds required to schedule a set of unsplittable flows of non-uniform sizes on a given path with non-uniform edge capacities. rufpp is NP-hard and constant-factor approximation algorithms are known under the no bottleneck assumption (NBA), which stipulates that maximum size of a flow is at most the minimum edge capacity. We study rufpp without the NBA, and present improved online and offline algorithms. We first study offline rufpp for a restricted class of instances called $\alpha$-small, where the size of each flow is at most $\alpha$ times the capacity of its bottleneck edge, and present an $O(\log(1/(1-\alpha)))$-approximation algorithm. Our main result is an online $O(\log\log c_{\max})$-competitive algorithm for rufpp for general instances, where $c_{\max}$ is the largest edge capacities, improving upon the previous best bound of $O(\log c_{\max})$ due to Epstein et al. Our result leads to an offline $O(\min(\log n, \log m, \log\log c_{\max}))$-approximation algorithm and an online $O(\min(\log m, \log\log c_{\max}))$-competitive algorithm for rufpp, where $n$ is the number of flows and $m$ is the number of edges.