IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v43y2022i5d10.1007_s10878-020-00646-5.html
   My bibliography  Save this article

An approximation algorithm for a general class of parametric optimization problems

Author

Listed:
  • Cristina Bazgan

    (Université Paris-Dauphine)

  • Arne Herzel

    (University of Kaiserslautern)

  • Stefan Ruzika

    (University of Kaiserslautern)

  • Clemens Thielen

    (Technical University of Munich)

  • Daniel Vanderpooten

    (Université Paris-Dauphine)

Abstract

In a (linear) parametric optimization problem, the objective value of each feasible solution is an affine function of a real-valued parameter and one is interested in computing a solution for each possible value of the parameter. For many important parametric optimization problems including the parametric versions of the shortest path problem, the assignment problem, and the minimum cost flow problem, however, the piecewise linear function mapping the parameter to the optimal objective value of the corresponding non-parametric instance (the optimal value function) can have super-polynomially many breakpoints (points of slope change). This implies that any optimal algorithm for such a problem must output a super-polynomial number of solutions. We provide a method for lifting approximation algorithms for non-parametric optimization problems to their parametric counterparts that is applicable to a general class of parametric optimization problems. The approximation guarantee achieved by this method for a parametric problem is arbitrarily close to the approximation guarantee of the algorithm for the corresponding non-parametric problem. It outputs polynomially many solutions and has polynomial running time if the non-parametric algorithm has polynomial running time. In the case that the non-parametric problem can be solved exactly in polynomial time or that an FPTAS is available, the method yields an FPTAS. In particular, under mild assumptions, we obtain the first parametric FPTAS for each of the specific problems mentioned above and a $$(3/2 + \varepsilon )$$ ( 3 / 2 + ε ) -approximation algorithm for the parametric metric traveling salesman problem. Moreover, we describe a post-processing procedure that, if the non-parametric problem can be solved exactly in polynomial time, further decreases the number of returned solutions such that the method outputs at most twice as many solutions as needed at minimum for achieving the desired approximation guarantee.

Suggested Citation

  • Cristina Bazgan & Arne Herzel & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2022. "An approximation algorithm for a general class of parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 43(5), pages 1328-1358, July.
    • Handle: RePEc:spr:jcomop:v:43:y:2022:i:5:d:10.1007_s10878-020-00646-5
    DOI: 10.1007/s10878-020-00646-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-020-00646-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-020-00646-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. S. Thomas McCormick, 1999. "Fast Algorithms for Parametric Scheduling Come From Extensions to Parametric Maximum Flow," Operations Research, INFORMS, vol. 47(5), pages 744-756, October.
    2. Hans Schneider & Michael H. Schneider, 1991. "Max-Balancing Weighted Directed Graphs and Matrix Scaling," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 208-222, February.
    3. Christos H. Papadimitriou & Mihalis Yannakakis, 1993. "The Traveling Salesman Problem with Distances One and Two," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 1-11, February.
    4. Mitsos, Alexander & Barton, Paul I., 2009. "Parametric mixed-integer 0-1 linear programming: The general case for a single parameter," European Journal of Operational Research, Elsevier, vol. 194(3), pages 663-686, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.
    2. E Duman & M H Ozcelik & A N Ceranoglu, 2005. "A TSP (1,2) application arising in cable assembly shops," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(6), pages 642-648, June.
    3. Zhang, Huili & Tong, Weitian & Xu, Yinfeng & Lin, Guohui, 2015. "The Steiner Traveling Salesman Problem with online edge blockages," European Journal of Operational Research, Elsevier, vol. 243(1), pages 30-40.
    4. Sedeno-Noda, A. & Alcaide, D. & Gonzalez-Martin, C., 2006. "Network flow approaches to pre-emptive open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1501-1518, November.
    5. Iosif Pappas & Nikolaos A. Diangelakis & Efstratios N. Pistikopoulos, 2021. "The exact solution of multiparametric quadratically constrained quadratic programming problems," Journal of Global Optimization, Springer, vol. 79(1), pages 59-85, January.
    6. Federico Della Croce, 2016. "MP or not MP: that is the question," Journal of Scheduling, Springer, vol. 19(1), pages 33-42, February.
    7. Martijn Ee & René Sitters, 2020. "The Chinese deliveryman problem," 4OR, Springer, vol. 18(3), pages 341-356, September.
    8. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2017. "Machine Speed Scaling by Adapting Methods for Convex Optimization with Submodular Constraints," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 724-736, November.
    9. Monnot, Jerome & Paschos, Vangelis Th. & Toulouse, Sophie, 2003. "Differential approximation results for the traveling salesman problem with distances 1 and 2," European Journal of Operational Research, Elsevier, vol. 145(3), pages 557-568, March.
    10. Shiming Li & Wei Yu & Zhaohui Liu, 2024. "Improved approximation algorithms for the k-path partition problem," Journal of Global Optimization, Springer, vol. 90(4), pages 983-1006, December.
    11. Bogdan Armaselu, 2023. "Approximation algorithms for some extensions of the maximum profit routing problem," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-22, January.
    12. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2020. "Scheduling problems with controllable processing times and a common deadline to minimize maximum compression cost," Journal of Global Optimization, Springer, vol. 76(3), pages 471-490, March.
    13. Klamroth, Kathrin & Wiecek, Margaret M., 2001. "A time-dependent multiple criteria single-machine scheduling problem," European Journal of Operational Research, Elsevier, vol. 135(1), pages 17-26, November.
    14. Mathias Weller & Annie Chateau & Rodolphe Giroudeau & Jean-Claude König & Valentin Pollet, 2018. "On residual approximation in solution extension problems," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1195-1220, November.
    15. Demange, Marc & Paschos, Vangelis Th., 2005. "Polynomial approximation algorithms with performance guarantees: An introduction-by-example," European Journal of Operational Research, Elsevier, vol. 165(3), pages 555-568, September.
    16. Maria Scutellà, 2007. "A note on the parametric maximum flow problem and some related reoptimization issues," Annals of Operations Research, Springer, vol. 150(1), pages 231-244, March.
    17. Sedeño-Noda, A. & de Pablo, D. Alcaide López & González-Martín, C., 2009. "A network flow-based method to solve performance cost and makespan open-shop scheduling problems with time-windows," European Journal of Operational Research, Elsevier, vol. 196(1), pages 140-154, July.
    18. Richard Oberdieck & Martina Wittmann-Hohlbein & Efstratios Pistikopoulos, 2014. "A branch and bound method for the solution of multiparametric mixed integer linear programming problems," Journal of Global Optimization, Springer, vol. 59(2), pages 527-543, July.
    19. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2016. "Application of Submodular Optimization to Single Machine Scheduling with Controllable Processing Times Subject to Release Dates and Deadlines," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 148-161, February.
    20. Wenkai Dai & Yongjie Yang, 2019. "Reoptimization of minimum latency problem revisited: don’t panic when asked to revisit the route after local modifications," Journal of Combinatorial Optimization, Springer, vol. 37(2), pages 601-619, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:43:y:2022:i:5:d:10.1007_s10878-020-00646-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.