Document Open Access Logo

Fixed-Parameter Algorithms for Unsplittable Flow Cover

Authors Andrés Cristi , Mathieu Mari, Andreas Wiese

PDF
Thumbnail PDF

File

LIPIcs.STACS.2020.42.pdf
  • Filesize: 0.54 MB
  • 17 pages

Document Identifiers

Author Details

Andrés Cristi
  • Universidad de Chile, Santiago, Chile
Mathieu Mari
  • École Normale Supérieure, Université PSL, Paris, France
Andreas Wiese
  • Universidad de Chile, Santiago, Chile

Funding

  • Cristi, Andrés: Supported by CONICYT-PFCHA/Doctorado Nacional/2018-21180347.
  • Mari, Mathieu: This work was partially funded by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR).
  • Wiese, Andreas: Partially supported by FONDECYT Regular grant 1170223.

Cite As Get BibTex

Andrés Cristi, Mathieu Mari, and Andreas Wiese. Fixed-Parameter Algorithms for Unsplittable Flow Cover. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.42

Abstract

The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem [Bar-Noy et al., STOC 2000] and also a QPTAS [Höhn et al., ICALP 2014]. In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slightly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of f(k)⋅ n^O_ε(1) that outputs a solution with at most (1+ε)k tasks or assert that there is no solution with at most k tasks. In this algorithm we use a new trick that intuitively allows us to pretend that we can select tasks from OPT multiple times.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Fixed parameter tractability
Keywords
  • Unsplittable Flow Cover
  • fixed parameter algorithms
  • approximation algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Susanne Albers, Sanjeev Arora, and Sanjeev Khanna. Page replacement for general caching problems. In SODA, volume 99, pages 31-40. Citeseer, 1999. Google Scholar
  2. Antonios Antoniadis, Ruben Hoeksma, Julie Meißner, José Verschae, and Andreas Wiese. A QPTAS for the General Scheduling Problem with Identical Release Dates. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.31.
  3. Nikhil Bansal, Amit Chakrabarti, Amir Epstein, and Baruch Schieber. A quasi-PTAS for unsplittable flow on line graphs. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC 2006), pages 721-729. ACM, 2006. Google Scholar
  4. Nikhil Bansal, Ravishankar Krishnaswamy, and Barna Saha. On capacitated set cover problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 38-49. Springer, 2011. Google Scholar
  5. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph (Seffi) Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. J. ACM, 48(5):1069-1090, 2001. URL: https://doi.org/10.1145/502102.502107.
  6. Jatin Batra, Naveen Garg, Amit Kumar, Tobias Mömke, and Andreas Wiese. New approximation schemes for unsplittable flow on a path. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 47-58, 2015. URL: https://doi.org/10.1137/1.9781611973730.5.
  7. Cristina Bazgan. Schémas d'approximation et Complexité Paramétrée, Rapport du stage (DEA). Technical report, Universitée Paris Sud, 1995. Google Scholar
  8. Laszlo A. Belady. A study of replacement algorithms for a virtual-storage computer. IBM Systems journal, 5(2):78-101, 1966. Google Scholar
  9. Paul Bonsma, Jens Schulz, and Andreas Wiese. A constant-factor approximation algorithm for unsplittable flow on paths. SIAM Journal on Computing, 43:767-799, 2014. Google Scholar
  10. Liming Cai and Xiuzhen Huang. Fixed-parameter approximation: Conceptual framework and approximability results. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 96-108, 2006. URL: https://doi.org/10.1007/11847250_9.
  11. Robert D Carr, Lisa K Fleischer, Vitus J Leung, and Cynthia A Phillips. Strengthening integrality gaps for capacitated network design and covering problems. In Proceedings of the 11th annual ACM-SIAM symposium on Discrete algorithms (SODA 2000), pages 106-115. Society for Industrial and Applied Mathematics, 2000. Google Scholar
  12. Marco Cesati and Luca Trevisan. On the efficiency of polynomial time approximation schemes. Inf. Process. Lett., 64(4):165-171, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00164-6.
  13. Deeparnab Chakrabarty, Elyot Grant, and Jochen Könemann. On column-restricted and priority covering integer programs. In International Conference on Integer Programming and Combinatorial Optimization, pages 355-368. Springer, 2010. Google Scholar
  14. Yijia Chen, Martin Grohe, and Magdalena Grüber. On parameterized approximability. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 109-120, 2006. URL: https://doi.org/10.1007/11847250_10.
  15. Maurice Cheung, Julián Mestre, David B Shmoys, and José Verschae. A primal-dual approximation algorithm for min-sum single-machine scheduling problems. SIAM Journal on Discrete Mathematics, 31(2):825-838, 2017. Google Scholar
  16. M. Chrobak, G. Woeginger, K. Makino, and H. Xu. Caching is hard, even in the fault model. In ESA, pages 195-206, 2010. Google Scholar
  17. Andrés Cristi and Andreas Wiese. Better approximations for general caching and UFP-cover under resource augmentation. Unpublished, 2019. Google Scholar
  18. Rodney G. Downey, Michael R. Fellows, and Catherine McCartin. Parameterized approximation problems. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 121-129, 2006. URL: https://doi.org/10.1007/11847250_11.
  19. Michael R Fellows and Neal Koblitz. Fixed-parameter complexity and cryptography. In International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pages 121-131. Springer, 1993. Google Scholar
  20. P. A. Franaszek and T. J. Wagner. Some distribution-free aspects of paging algorithm performance. J. ACM, 21(1):31-39, January 1974. URL: https://doi.org/10.1145/321796.321800.
  21. Fabrizio Grandoni, Tobias Mömke, Andreas Wiese, and Hang Zhou. To augment or not to augment: Solving unsplittable flow on a path by creating slack. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), 2017. To appear. Google Scholar
  22. Fabrizio Grandoni, Tobias Mömke, Andreas Wiese, and Hang Zhou. A (5/3+ε)-approximation for unsplittable flow on a path: placing small tasks into boxes. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2018), pages 607-619. ACM, 2018. Google Scholar
  23. Wiebke Höhn, Julián Mestre, and Andreas Wiese. How unsplittable-flow-covering helps scheduling with job-dependent cost functions. Algorithmica, 80(4):1191-1213, 2018. Google Scholar
  24. Sandy Irani. Page replacement with multi-size pages and applications to web caching. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 701-710. ACM, 1997. Google Scholar
  25. Daniel Lokshtanov, Fahad Panolan, MS Ramanujan, and Saket Saurabh. Lossy kernelization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2017), pages 224-237. ACM, 2017. Google Scholar
  26. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008. URL: https://doi.org/10.1093/comjnl/bxm048.
  27. Andreas Wiese. A (1+epsilon)-approximation for unsplittable flow on a path in fixed-parameter running time. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pages 67:1-67:13, 2017. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact