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Courcelle’s Theorem for Lipschitz Continuity

Authors Tatsuya Gima , Soh Kumabe , Yuichi Yoshida

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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Fixed-Parameter Tractability
  • Algorithmic Meta-Theorem
  • Lipschitz Continuity

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Abstract

Lipschitz continuity of algorithms, introduced by Kumabe and Yoshida (FOCS'23), measures the stability of an algorithm against small input perturbations. Algorithms with small Lipschitz continuity are desirable, as they ensure reliable decision-making and reproducible scientific research. Several studies have proposed Lipschitz continuous algorithms for various combinatorial optimization problems, but these algorithms are problem-specific, requiring a separate design for each problem.
To address this issue, we provide the first algorithmic meta-theorem in the field of Lipschitz continuous algorithms. Our result can be seen as a Lipschitz continuous analogue of Courcelle’s theorem, which offers Lipschitz continuous algorithms for problems on bounded-treewidth graphs. Specifically, we consider the problem of finding a vertex set in a graph that maximizes or minimizes the total weight, subject to constraints expressed in monadic second-order logic (MSO₂). We show that for any ε > 0, there exists a (1±ε)-approximation algorithm for the problem with a polylogarithmic Lipschitz constant on bounded treewidth graphs. On such graphs, our result outperforms most existing Lipschitz continuous algorithms in terms of approximability and/or Lipschitz continuity. Further, we provide similar results for problems on bounded-clique-width graphs subject to constraints expressed in MSO₁. Additionally, we construct a Lipschitz continuous version of Baker’s decomposition using our meta-theorem as a subroutine.

Cite As Get BibTex

Tatsuya Gima, Soh Kumabe, and Yuichi Yoshida. Courcelle’s Theorem for Lipschitz Continuity. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.11

Author Details

Tatsuya Gima
  • Hokkaido University, Sapporo, Japan
Soh Kumabe
  • CyberAgent, Tokyo, Japan
Yuichi Yoshida
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Gima, Tatsuya: JSPS KAKENHI Grant Numbers JP24K23847, JP25K03077
  • Yoshida, Yuichi: JSPS KAKENHI Grant Number JP24K02903

References

  1. Brenda Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM (JACM), 41(1):153-180, 1994. URL: https://doi.org/10.1145/174644.174650.
  2. Hans Bodlaender and Torben Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. SIAM Journal on Computing, 27(6):1725-1746, 1998. URL: https://doi.org/10.1137/S0097539795289859.
  3. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  4. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012. URL: https://doi.org/10.1017/CBO9780511977619.
  5. Bruno Courcelle and Mohamed Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science, 109(1&2):49-82, 1993. URL: https://doi.org/10.1016/0304-3975(93)90064-Z.
  6. Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity classifications of Boolean constraint satisfaction problems. SIAM, 2001. Google Scholar
  7. Cynthia Dwork. Differential privacy. In International Colloquium on Automata, Languages and Programming (ICALP), volume 4052, pages 1-12, 2006. URL: https://doi.org/10.1007/11787006_1.
  8. Noah Fleming and Yuichi Yoshida. Sensitivity lower bounds for approximation algorithms. arXiv preprint arXiv:2411.02744, 2024. URL: https://doi.org/10.48550/arXiv.2411.02744.
  9. Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David Williamson. The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863-1920, 2001. URL: https://doi.org/10.1137/S0097539799349948.
  10. Tuukka Korhonen. Lower bounds on dynamic programming for maximum weight independent set. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, International Colloquium on Automata, Languages, and Programming (ICALP), volume 198 of LIPIcs, pages 87:1-87:14, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.87.
  11. Stephan Kreutzer. Algorithmic meta-theorems. In Finite and Algorithmic Model Theory, volume 379 of London Mathematical Society Lecture Note Series, pages 177-270. Cambridge University Press, 2011. Google Scholar
  12. Soh Kumabe and Yuichi Yoshida. Average sensitivity of dynamic programming. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1925-1961, 2022. URL: https://doi.org/10.1137/1.9781611977073.77.
  13. Soh Kumabe and Yuichi Yoshida. Average sensitivity of the knapsack problem. In European Symposium on Algorithms (ESA), volume 244, pages 75:1-75:14, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.75.
  14. Soh Kumabe and Yuichi Yoshida. Lipschitz continuous algorithms for graph problems. In Symposium on Foundations of Computer Science (FOCS), pages 762-797, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00051.
  15. Soh Kumabe and Yuichi Yoshida. Lipschitz continuous allocations for optimization games. In International Colloquium on Automata, Languages, and Programming (ICALP), volume 297, pages 102:1-102:16, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.102.
  16. Soh Kumabe and Yuichi Yoshida. Lipschitz continuous algorithms for covering problems. In Symposium on Discrete Algorithms (SODA), pages 1057-1093, 2025. URL: https://doi.org/10.1137/1.9781611978322.31.
  17. Leonid Libkin. Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2004. URL: https://doi.org/10.1007/978-3-662-07003-1.
  18. Quanquan Liu, Grigoris Velegkas, Yuichi Yoshida, and Felix Zhou. Pointwise lipschitz continuous graph algorithms via proximal gradient analysis. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2405.08938.
  19. Shogo Murai and Yuichi Yoshida. Sensitivity analysis of centralities on unweighted networks. In The world wide web conference, pages 1332-1342, 2019. URL: https://doi.org/10.1145/3308558.3313422.
  20. Pan Peng and Yuichi Yoshida. Average sensitivity of spectral clustering. In ACM SIGKDD international conference on knowledge discovery & data mining (KDD), pages 1132-1140, 2020. URL: https://doi.org/10.1145/3394486.3403166.
  21. Nithin Varma and Yuichi Yoshida. Average sensitivity of graph algorithms. SIAM Journal on Computing, 52(4):1039-1081, 2023. URL: https://doi.org/10.1137/21M1399592.
  22. Yuichi Yoshida and Shinji Ito. Average sensitivity of euclidean k-clustering. Advances in Neural Information Processing Systems (NeurIPS), 35:32487-32498, 2022. Google Scholar
  23. Yuichi Yoshida and Samson Zhou. Sensitivity analysis of the maximum matching problem. In Innovations in Theoretical Computer Science Conference (ITCS), volume 185, pages 58:1-58:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.58.
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