Document Open Access Logo

Resource Sharing Revisited: Local Weak Duality and Optimal Convergence

Author Daniel Blankenburg

PDF
Thumbnail PDF

File

LIPIcs.ESA.2022.20.pdf
  • Filesize: 0.89 MB
  • 14 pages

Document Identifiers

Author Details

Daniel Blankenburg
  • Research Institute for Discrete Mathematics, Universität Bonn, Germany

Acknowledgements

We thank Jens Vygen for many fruitful discussions and Sebastian Pokutta for helpful suggestions.

Cite AsGet BibTex

Daniel Blankenburg. Resource Sharing Revisited: Local Weak Duality and Optimal Convergence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.20

Abstract

We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an 𝓁_∞-minimal element in a Minkowski sum 𝒳 = ∑_{C ∈ 𝒞} X_C of non-empty closed convex sets X_C ⊆ ℝ^ℛ_{≥ 0}, where 𝒞 and ℛ are finite sets. We assume that an oracle for approximate linear minimization over X_C is given. We improve on the currently fastest known FPTAS in various ways. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately 𝓁_{∞}-minimal, but among such solutions, also its second-highest entry is approximately minimal. Based on a result by Klein and Young [Klein and Young, 2015], we provide a lower bound of Ω((𝒞|+|ℛ|)/δ² log |ℛ|) required oracle calls for a natural class of algorithms. Our FPTAS is optimal within this class - its running time matches the lower bound precisely, and thus improves on the previously best-known running time for the primal as well as the dual problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Resource sharing
  • Dantzig-Wolfe-type algorithms
  • Decreasing minimization

Metrics

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

Related Versions

References

  1. Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly linear-time packing and covering lp solvers: Achieving width-independence and -convergence. Mathematical Programming, 175, February 2018. URL: https://doi.org/10.1007/s10107-018-1244-x.
  2. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(6):121-164, 2012. URL: https://doi.org/10.4086/toc.2012.v008a006.
  3. Daniel Bienstock and Garud Iyengar. Approximating fractional packings and coverings in o(1/epsilon) iterations. SIAM J. Comput., 35:825-854, 2006. Google Scholar
  4. Bhaskar Dutta and Debraj Ray. A concept of egalitarianism under participation constraints. Econometrica, 57:615-35, February 1989. URL: https://doi.org/10.2307/1911055.
  5. Lisa K. Fleischer. Approximating fractional multicommodity flow independent of the number of commodities. SIAM Journal on Discrete Mathematics, 13(4):505-520, 2000. URL: https://doi.org/10.1137/S0895480199355754.
  6. András Frank and Kazuo Murota. Discrete decreasing minimization, part i: Base-polyhedra with applications in network optimization, 2019. URL: http://arxiv.org/abs/1808.07600.
  7. András Frank and Kazuo Murota. Discrete decreasing minimization, part ii: Views from discrete convex analysis, 2020. URL: http://arxiv.org/abs/1808.08477.
  8. András Frank and Kazuo Murota. Fair integral flows, 2020. URL: http://arxiv.org/abs/1907.02673.
  9. Marguerite Frank and Philip Wolfe. An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3(1-2):95-110, 1956. URL: https://doi.org/10.1002/nav.3800030109.
  10. Satoru Fujishige. Lexicographically optimal base of a polymatroid with respect to a weight vector. Mathematics of Operations Research, 5(2):186-196, 1980. URL: http://www.jstor.org/stable/3689149.
  11. Naveen Garg and Jochen Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM J. Comput., 37:630-652, January 2007. URL: https://doi.org/10.1109/SFCS.1998.743463.
  12. Michael Gester, Dirk Müller, Tim Nieberg, Christian Panten, Christian Schulte, and Jens Vygen. Bonnroute: Algorithms and data structures for fast and good vlsi routing. ACM Trans. Des. Autom. Electron. Syst., 18(2), April 2013. URL: https://doi.org/10.1145/2442087.2442103.
  13. M. Grigoriadis and L. Khachiyan. Fast approximation schemes for convex programs with many blocks and coupling constraints. SIAM J. Optim., 4:86-107, 1994. Google Scholar
  14. Michael D. Grigoriadis and Leonid G. Khachiyan. Coordination complexity of parallel price-directive decomposition. Mathematics of Operations Research, 21(2):321-340, 1996. URL: https://doi.org/10.1287/moor.21.2.321.
  15. G. H. Hardy, George Polya, and J. E. Littlewood. Inequalities. The University press Cambridge [Eng.], 1934. Google Scholar
  16. Stephan Held, Dirk Müller, Daniel Rotter, Rudolf Scheifele, Vera Traub, and Jens Vygen. Global routing with timing constraints. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 37(2):406-419, 2018. URL: https://doi.org/10.1109/TCAD.2017.2697964.
  17. Martin Jaggi. Revisiting Frank-Wolfe: Projection-free sparse convex optimization. In Sanjoy Dasgupta and David McAllester, editors, Proceedings of the 30th International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pages 427-435, Atlanta, Georgia, USA, 17-19 June 2013. PMLR. URL: http://proceedings.mlr.press/v28/jaggi13.html.
  18. Klaus Jansen and Hu Zhang. Approximation algorithms for general packing problems and their application to the multicast congestion problem. Mathematical Programming, 114:183-206, 2008. Google Scholar
  19. Thomas Kerdreux. Accelerating conditional gradient methods. PhD thesis, Université Paris sciences et lettres, June 2020. Google Scholar
  20. Rohit Khandekar. Lagrangian relaxation based algorithms for convex programming problems. PhD thesis, Indian Institute of Technology Delhi, 2004. Google Scholar
  21. Philip Klein and Neal E. Young. On the number of iterations for dantzig-wolfe optimization and packing-covering approximation algorithms. SIAM Journal on Computing, 44(4):1154-1172, 2015. URL: https://doi.org/10.1137/12087222X.
  22. Christos Koufogiannakis and Neal E. Young. A nearly linear-time ptas for explicit fractional packing and covering linear programs. Algorithmica, 70(4):648-674, 2014. Journal version of [2007]. URL: https://doi.org/10.1007/s00453-013-9771-6.
  23. Nimrod Megiddo. A good algorithm for lexicographically optimal flows in multi-terminal networks. Bulletin of the American Mathematical Society, 83:407-409, 1977. Google Scholar
  24. Dirk Müller, Klaus Radke, and Jens Vygen. Faster min–max resource sharing in theory and practice. Mathematical Programming Computation, 3:1-35, 2011. Google Scholar
  25. Dritan Nace and James Orlin. Lexicographically minimum and maximum load linear programming problems. Operations Research, 55:182-187, February 2007. URL: https://doi.org/10.1287/opre.1060.0341.
  26. Serge Plotkin, David Shmoys, and Éva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20, January 2001. URL: https://doi.org/10.1109/SFCS.1991.185411.
  27. Bozidar Radunovic and Jean-Yves Le Boudec. A unified framework for max-min and min-max fairness with applications. IEEE/ACM Transactions on Networking, 15(5):1073-1083, 2007. URL: https://doi.org/10.1109/TNET.2007.896231.
  28. Prabhakar Raghavan and Clark D. Tompson. Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365-374, December 1987. URL: https://doi.org/10.1007/BF02579324.
  29. Farhad Shahrokhi and David W. Matula. The maximum concurrent flow problem. J. ACM, 37(2):318-334, April 1990. URL: https://doi.org/10.1145/77600.77620.
  30. Arie Tamir. Least majorized elements and generalized polymatroids. Mathematics of Operations Research, 20(3):583-589, 1995. URL: https://doi.org/10.1287/moor.20.3.583.
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 Imprint Privacy Contact