TY - GEN
T1 - Unsplittable Flow on a Short Path
AU - Doron-Arad, Ilan
AU - Grandoni, Fabrizio
AU - Kulik, Ariel
N1 - Publisher Copyright:
© Ilan Doron-Arad, Fabrizio Grandoni, and Ariel Kulik.
PY - 2024/12/5
Y1 - 2024/12/5
N2 - 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.
AB - 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.
KW - Approximation Schemes
KW - Knapsack
KW - Parameterized Approximations
UR - http://www.scopus.com/inward/record.url?scp=85213380903&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.IPEC.2024.5
DO - 10.4230/LIPIcs.IPEC.2024.5
M3 - Conference contribution
AN - SCOPUS:85213380903
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 19th International Symposium on Parameterized and Exact Computation, IPEC 2024
A2 - Bonnet, Edouard
A2 - Rzazewski, Pawel
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 19th International Symposium on Parameterized and Exact Computation, IPEC 2024
Y2 - 4 September 2024 through 6 September 2024
ER -