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A logarithmic approximation for unsplittable flow on line graphs

Authors:

Zachary Friggstad,

Rohit Khandekar,

Mohammad R. SalavatipourAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 10, Issue 1

Article No.: 1, Pages 1 - 15

https://doi.org/10.1145/2532645

Published: 01 January 2014 Publication History

Abstract

We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time s_i, an end time t_i, a demand d_i > 0, and a profit p_i > 0. A task, if accepted, requires d_i units of “bandwidth” from time s_i to t_i and accrues a profit of p_i. For every time t, we are also specified the available bandwidth c_t, and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints.

We present the first polynomial time O(log n) approximation algorithm for this problem. This significantly advances the state of the art, as no polynomial time o(n) approximation was known previously. Previous results for this problem were known only in more restrictive settings; in particular, either the instance satisfies the so-called “no-bottleneck” assumption: max_i d_i ≤ min_t c_t, or the ratio of both maximum to minimum demands and maximum to minimum capacities are polynomially (or quasi-polynomially) bounded in n. Our result, on the other hand, does not require these assumptions.

Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an Ω(n) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming.

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Cited By

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https://dl.acm.org/doi/10.1007/s10951-023-00786-6
Karapetyan AElbassioni KKhonji MChau S(2022)Approximations for generalized unsplittable flow on paths with application to power systems optimizationAnnals of Operations Research10.1007/s10479-022-05054-y320:1(173-204)Online publication date: 7-Nov-2022
https://doi.org/10.1007/s10479-022-05054-y
Hu XKoutris PBlanas SLibkin LPichler RGuagliardo P(2021)Algorithms for a Topology-aware Massively Parallel Computation ModelProceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3452021.3458318(199-214)Online publication date: 20-Jun-2021
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Index Terms

A logarithmic approximation for unsplittable flow on line graphs
1. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 10, Issue 1

January 2014

73 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2578701

Issue’s Table of Contents

Copyright © 2014 ACM.

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Association for Computing Machinery

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Publication History

Published: 01 January 2014

Accepted: 01 June 2009

Revised: 01 March 2009

Received: 01 February 2007

Published in TALG Volume 10, Issue 1

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Cited By

Sarpatwar KSchieber BShachnai H(2024)The preemptive resource allocation problemJournal of Scheduling10.1007/s10951-023-00786-627:1(103-118)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s10951-023-00786-6
Karapetyan AElbassioni KKhonji MChau S(2022)Approximations for generalized unsplittable flow on paths with application to power systems optimizationAnnals of Operations Research10.1007/s10479-022-05054-y320:1(173-204)Online publication date: 7-Nov-2022
https://doi.org/10.1007/s10479-022-05054-y
Hu XKoutris PBlanas SLibkin LPichler RGuagliardo P(2021)Algorithms for a Topology-aware Massively Parallel Computation ModelProceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3452021.3458318(199-214)Online publication date: 20-Jun-2021
https://dl.acm.org/doi/10.1145/3452021.3458318
Brubach BSankararaman KSrinivasan AXu P(2019)Algorithms to Approximate Column-sparse Packing ProblemsACM Transactions on Algorithms10.1145/335540016:1(1-32)Online publication date: 15-Nov-2019
https://dl.acm.org/doi/10.1145/3355400
Brubach BSankararaman KSrinivasan AXu PCzumaj A(2018)Algorithms to approximate column-sparse packing problemsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175289(311-330)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175289
Sarpatwar KSchieber BShachnai HScheideler CFineman J(2018)Brief AnnouncementProceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures10.1145/3210377.3210666(343-345)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3210377.3210666
Bar-Yehuda RBeder MRawitz D(2017)A Constant Factor Approximation Algorithm for the Storage Allocation ProblemAlgorithmica10.1007/s00453-016-0137-877:4(1105-1127)Online publication date: 1-Apr-2017
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Li WLi JGuan LShi Y(2015)The Prize-collecting Call Control Problem on Weighted Lines and RingsRAIRO - Operations Research10.1051/ro/201501050:1(39-46)Online publication date: 26-Aug-2015
https://doi.org/10.1051/ro/2015010

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