article

Using fractional primal-dual to schedule split intervals with demands

Authors:

Reuven Bar-Yehuda,

Dror RawitzAuthors Info & Claims

Discrete Optimization, Volume 3, Issue 4

Pages 275 - 287

https://doi.org/10.1016/j.disopt.2006.05.010

Published: 01 December 2006 Publication History

Abstract

We consider the problem of scheduling jobs that are given as groups of non-intersecting intervals on the real line. Each job j is associated with a t-interval (which consists of up to t segments, for some t>=1), a demand, d"j@?[0,1], and a weight, w(j). A feasible schedule is a collection of jobs such that, for every s@?R, the total demand of the jobs in the schedule whose t-interval contains s does not exceed 1. Our goal is to find a feasible schedule that maximizes the total weight of scheduled jobs. We present a 6t-approximation algorithm for this problem that uses a novel extension of the primal-dual schema called fractional primal-dual. The first step in a fractional primal-dual r-approximation algorithm is to compute an optimal solution, x^*, of an LP relaxation of the problem. Next, the algorithm produces an integral primal solution x, and a new LP, denoted by P', that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. Moreover, x^* is a feasible solution of P'. The algorithm also computes a solution y to the dual of P'. The solution x is r-approximate, since its weight is bounded by the value of y divided by r. We present a fractional local ratio interpretation of our 6t-approximation algorithm. We also discuss the connection between fractional primal-dual and the fractional local ratio technique. Specifically, we show that the former is the primal-dual manifestation of the latter.

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Cited By

Halldórsson MRawitz D(2024)Distributed Fractional Local Ratio and Independent Set ApproximationStructural Information and Communication Complexity10.1007/978-3-031-60603-8_16(281-299)Online publication date: 27-May-2024
https://dl.acm.org/doi/10.1007/978-3-031-60603-8_16
Wan PYuan HWang JRen JZhang Y(2018)Joint Selection and Scheduling of Communication Requests in Multi-Channel Wireless Networks under SINR ModelIEEE INFOCOM 2018 - IEEE Conference on Computer Communications10.1109/INFOCOM.2018.8486389(2312-2320)Online publication date: 16-Apr-2018
https://dl.acm.org/doi/10.1109/INFOCOM.2018.8486389
Wan PDressler Fauf der Heide F(2016)Joint selection and transmission scheduling of point-to-point communication requests in multi-channel wireless networksProceedings of the 17th ACM International Symposium on Mobile Ad Hoc Networking and Computing10.1145/2942358.2942378(231-240)Online publication date: 5-Jul-2016
https://dl.acm.org/doi/10.1145/2942358.2942378
Show More Cited By

Using fractional primal-dual to schedule split intervals with demands
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
    2. Mathematical optimization
      1. Continuous optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
    2. Online algorithms
      1. Online learning algorithms
  2. Theory and algorithms for application domains
    1. Machine learning theory
      1. Reinforcement learning

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cover image Discrete Optimization

Discrete Optimization Volume 3, Issue 4

December, 2006

128 pages

ISSN:1572-5286

Issue’s Table of Contents

Copyright © Elsevier Ltd © 2006.

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 December 2006

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Cited By

Halldórsson MRawitz D(2024)Distributed Fractional Local Ratio and Independent Set ApproximationStructural Information and Communication Complexity10.1007/978-3-031-60603-8_16(281-299)Online publication date: 27-May-2024
https://dl.acm.org/doi/10.1007/978-3-031-60603-8_16
Wan PYuan HWang JRen JZhang Y(2018)Joint Selection and Scheduling of Communication Requests in Multi-Channel Wireless Networks under SINR ModelIEEE INFOCOM 2018 - IEEE Conference on Computer Communications10.1109/INFOCOM.2018.8486389(2312-2320)Online publication date: 16-Apr-2018
https://dl.acm.org/doi/10.1109/INFOCOM.2018.8486389
Wan PDressler Fauf der Heide F(2016)Joint selection and transmission scheduling of point-to-point communication requests in multi-channel wireless networksProceedings of the 17th ACM International Symposium on Mobile Ad Hoc Networking and Computing10.1145/2942358.2942378(231-240)Online publication date: 5-Jul-2016
https://dl.acm.org/doi/10.1145/2942358.2942378
Fomin FGaspers SGolovach PSuchan KSzeider Svan Leeuwen EVatshelle MVillanger Y(2012)k-gap interval graphsProceedings of the 10th Latin American international conference on Theoretical Informatics10.1007/978-3-642-29344-3_30(350-361)Online publication date: 16-Apr-2012
https://dl.acm.org/doi/10.1007/978-3-642-29344-3_30
Borodin ACashman DMagen A(2011)How well can primal-dual and local-ratio algorithms perform?ACM Transactions on Algorithms10.1145/1978782.19787847:3(1-26)Online publication date: 15-Jul-2011
https://dl.acm.org/doi/10.1145/1978782.1978784
Bachmann UHalldórsson MShachnai H(2010)Online selection of intervals and t-intervalsProceedings of the 12th Scandinavian conference on Algorithm Theory10.1007/978-3-642-13731-0_36(383-394)Online publication date: 21-Jun-2010
https://dl.acm.org/doi/10.1007/978-3-642-13731-0_36
Rawitz D(2007)Admission control with advance reservations in simple networksJournal of Discrete Algorithms10.1016/j.jda.2006.10.0085:3(491-500)Online publication date: 1-Sep-2007
https://dl.acm.org/doi/10.1016/j.jda.2006.10.008

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