Document Open Access Logo

Parameterized Complexity of Submodular Minimization Under Uncertainty

Authors Naonori Kakimura , Ildikó Schlotter

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2024.30.pdf
  • Filesize: 0.85 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Submodular minimization
  • optimization under uncertainty
  • parameterized complexity
  • cut function

Metrics

Abstract

This paper studies the computational complexity of a robust variant of a two-stage submodular minimization problem that we call Robust Submodular Minimizer. In this problem, we are given k submodular functions f_1,… ,f_k over a set family 2^V, which represent k possible scenarios in the future when we will need to find an optimal solution for one of these scenarios, i.e., a minimizer for one of the functions. The present task is to find a set X ⊆ V that is close to some optimal solution for each f_i in the sense that some minimizer of f_i can be obtained from X by adding/removing at most d elements for a given integer d ∈ ℕ. The main contribution of this paper is to provide a complete computational map of this problem with respect to parameters k and d, which reveals a tight complexity threshold for both parameters:  
- Robust Submodular Minimizer can be solved in polynomial time when k ≤ 2, but is NP-hard if k is a constant with k ≥ 3. 
- Robust Submodular Minimizer can be solved in polynomial time when d = 0, but is NP-hard if d is a constant with d ≥ 1. 
- Robust Submodular Minimizer is fixed-parameter tractable when parameterized by (k,d).  We also show that if some submodular function f_i has a polynomial number of minimizers, then the problem becomes fixed-parameter tractable when parameterized by d. We remark that all our hardness results hold even if each submodular function is given by a cut function of a directed graph.

Cite As Get BibTex

Naonori Kakimura and Ildikó Schlotter. Parameterized Complexity of Submodular Minimization Under Uncertainty. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.30

Author Details

Naonori Kakimura
  • Department of Mathematics, Keio University, Yokohama, Japan
Ildikó Schlotter
  • HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary
  • Budapest University of Technology and Economics, Hungary

Funding

  • Kakimura, Naonori: Supported by JSPS KAKENHI Grant Numbers JP22H05001, JP20H05795, and JP21H03397, Japan and JST ERATO Grant Number JPMJER2301, Japan.
  • Schlotter, Ildikó: Supported by the Hungarian Academy of Sciences under its János Bolyai Research Scholarship and its Momentum Programme (LP2021-2).

Acknowledgements

The authors are grateful to the organizers of the 14th Emléktábla Workshop which took place in Vác, Hungary in 2023, and provided a great opportunity for our initial discussions.

References

  1. Garrett Birkhoff. Rings of sets. Duke Math. J., 3(3):443-454, 1937. URL: https://doi.org/10.1215/S0012-7094-37-00334-X.
  2. Hans-Joachim Böckenhauer, Karin Freiermuth, Juraj Hromkovic, Tobias Mömke, Andreas Sprock, and Björn Steffen. Steiner tree reoptimization in graphs with sharpened triangle inequality. J. Discrete Algorithms, 11:73-86, 2012. URL: https://doi.org/10.1016/J.JDA.2011.03.014.
  3. Paul Bonsma. Most balanced minimum cuts. Discrete Applied Mathematics, 158:261-276, 2010. URL: https://doi.org/10.1016/j.dam.2009.09.010.
  4. Nicolas Boria and Vangelis Th. Paschos. Fast reoptimization for the minimum spanning tree problem. J. Discrete Algorithms, 8(3):296-310, 2010. URL: https://doi.org/10.1016/J.JDA.2009.07.002.
  5. Christina Büsing. Recoverable robust shortest path problems. Networks, 59(1):181-189, 2012. URL: https://doi.org/10.1002/NET.20487.
  6. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Almost-linear-time algorithms for maximum flow and minimum-cost flow. Commun. ACM, 66(12):85-92, 2023. URL: https://doi.org/10.1145/3610940.
  7. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Mitre Costa Dourado, Dirk Meierling, Lucia Draque Penso, Dieter Rautenbach, Fábio Protti, and Aline Ribeiro de Almeida. Robust recoverable perfect matchings. Networks, 66(3):210-213, 2015. URL: https://doi.org/10.1002/NET.21624.
  9. Rod G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  10. Dennis Fischer, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. An investigation of the recoverable robust assignment problem. In Petr A. Golovach and Meirav Zehavi, editors, 16th International Symposium on Parameterized and Exact Computation, IPEC 2021, September 8-10, 2021, Lisbon, Portugal, volume 214 of LIPIcs, pages 19:1-19:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.IPEC.2021.19.
  11. Moti Frances and Ami Litman. On covering problems of codes. Theory Comput. Syst., 30(2):113-119, 1997. URL: https://doi.org/10.1007/s002240000044.
  12. Marc Goerigk, Stefan Lendl, and Lasse Wulf. On the recoverable traveling salesman problem. CoRR, abs/2111.09691, 2021. URL: https://arxiv.org/abs/2111.09691.
  13. Felix Hommelsheim, Nicole Megow, Komal Muluk, and Britta Peis. Recoverable robust optimization with commitment. CoRR, abs/2306.08546, 2023. URL: https://arxiv.org/abs/2306.08546.
  14. Mikita Hradovich, Adam Kasperski, and Pawel Zielinski. Recoverable robust spanning tree problem under interval uncertainty representations. J. Comb. Optim., 34(2):554-573, 2017. URL: https://doi.org/10.1007/S10878-016-0089-6.
  15. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. A parameterized view to the robust recoverable base problem of matroids under structural uncertainty. Oper. Res. Lett., 50(3):370-375, 2022. URL: https://doi.org/10.1016/J.ORL.2022.05.001.
  16. Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Submodular reassignment problem for reallocating agents to tasks with synergy effects. Discret. Optim., 44(Part):100631, 2022. URL: https://doi.org/10.1016/j.disopt.2021.100631.
  17. Naonori Kakimura and Ildikó Schlotter. Parameterized complexity of submodular minimization under uncertainty. CoRR, abs/2404.07516, 2024. URL: https://arxiv.org/abs/2404.07516.
  18. Stefan Kratsch, Shaohua Li, Dániel Marx, Marcin Pilipczuk, and Magnus Wahlström. Multi-budgeted directed cuts. Algorithmica, 82(8):2135-2155, 2020. URL: https://doi.org/10.1007/S00453-019-00609-1.
  19. Thomas Lachmann, Stefan Lendl, and Gerhard J. Woeginger. A linear time algorithm for the robust recoverable selection problem. Discret. Appl. Math., 303:94-107, 2021. URL: https://doi.org/10.1016/J.DAM.2020.08.012.
  20. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1049-1065, 2015. URL: https://doi.org/10.1109/FOCS.2015.68.
  21. Stefan Lendl, Britta Peis, and Veerle Timmermans. Matroid bases with cardinality constraints on the intersection. Math. Program., 194(1):661-684, 2022. URL: https://doi.org/10.1007/S10107-021-01642-1.
  22. Christian Liebchen, Marco E. Lübbecke, Rolf H. Möhring, and Sebastian Stiller. The concept of recoverable robustness, linear programming recovery, and railway applications. In Ravindra K. Ahuja, Rolf H. Möhring, and Christos D. Zaroliagis, editors, Robust and Online Large-Scale Optimization: Models and Techniques for Transportation Systems, volume 5868 of Lecture Notes in Computer Science, pages 1-27. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-05465-5_1.
  23. Jérôme Monnot. A note on the traveling salesman reoptimization problem under vertex insertion. Inf. Process. Lett., 115(3):435-438, 2015. URL: https://doi.org/10.1016/J.IPL.2014.11.003.
  24. Kazuo Murota. Discrete Convex Analysis. SIAM, 2003. URL: https://doi.org/10.1137/1.9780898718508.
  25. James B. Orlin. A faster strongly polynomial time algorithm for submodular function minimization. Math. Program., 118(2):237-251, 2009. URL: https://doi.org/10.1007/s10107-007-0189-2.
  26. Jean-Claude Picard and Maurice Queyranne. On the structure of all minimum cuts in a network and applications. Mathematical Programming Studies, 13:8-16, 1980. URL: https://doi.org/10.1007/BFb0120902.
  27. Thomas J Schaefer. The complexity of satisfiability problems. In Proceedings of the Tenth ACM Symposium on Theory of Computing (STOC '78), pages 216-226. ACM, 1978. URL: https://doi.org/10.1145/800133.804350.
  28. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, 2003. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact