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Unsplittable Flow on a Short Path

Authors Ilan Doron-Arad, Fabrizio Grandoni, Ariel Kulik

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Author Details

Ilan Doron-Arad
  • Computer Science Department, Technion, Haifa, Israel
Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Ariel Kulik
  • Computer Science Department, Technion, Haifa, Israel

Funding

  • Grandoni, Fabrizio: Partially supported by the SNSF Grant 200021_2000731/1.
  • Kulik, Ariel: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 852780-ERC (SUBMODULAR).

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Ilan Doron-Arad, Fabrizio Grandoni, and Ariel Kulik. Unsplittable Flow on a Short Path. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.5

Abstract

In the Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities and a collection of tasks. Each task is characterized by a demand, a profit, and a subpath. Our goal is to select a maximum profit subset of tasks such that the total demand of the selected tasks that use each edge e is at most the capacity of e. BagUFP is the generalization of UFP where tasks are partitioned into bags, and we are allowed to select at most one task per bag. UFP admits a PTAS [Grandoni,Mömke,Wiese'22] but not an EPTAS [Wiese'17]. BagUFP is APX-hard [Spieksma'99] and the current best approximation is O(log n/log log n) [Grandoni,Ingala,Uniyal'15], where n is the number of tasks. 
In this paper, we study the mentioned two problems when parameterized by the number m of edges in the graph, with the goal of designing faster parameterized approximation algorithms. We present a parameterized EPTAS for BagUFP, and a substantially faster parameterized EPTAS for UFP (which is an FPTAS for m = O(1)). We also show that a parameterized FPTAS for UFP (hence for BagUFP) does not exist, therefore our results are qualitatively tight.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Knapsack
  • Approximation Schemes
  • Parameterized Approximations

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