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### Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back

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### Abstract

Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss.

### BibTeX - Entry

@InProceedings{grandoni_et_al:LIPIcs.ESA.2021.49,
author =	{Grandoni, Fabrizio and M\"{o}mke, Tobias and Wiese, Andreas},
title =	{{Faster (1+\epsilon)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back}},
booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
pages =	{49:1--49:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-204-4},
ISSN =	{1868-8969},
year =	{2021},
volume =	{204},
editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}