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The Descriptive Complexity of Relation Modification Problems

Authors Florian Chudigiewitsch , Marlene Gründel , Christian Komusiewicz , Nils Morawietz , Till Tantau

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ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Circuit complexity
  • Theory of computation → Problems, reductions and completeness
Keywords
  • graph problems
  • descriptive complexity
  • edge modification
  • parameterized complexity
  • circuit complexity

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Abstract

A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns. 
Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in para-AC^{0↑} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

Cite As Get BibTex

Florian Chudigiewitsch, Marlene Gründel, Christian Komusiewicz, Nils Morawietz, and Till Tantau. The Descriptive Complexity of Relation Modification Problems. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 93:1-93:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.93

Author Details

Florian Chudigiewitsch
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Marlene Gründel
  • Algorithms and Complexity Group, TU Wien, Austria
Christian Komusiewicz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
  • LaBRI, Université de Bordeaux, Talence, France
Till Tantau
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany

Funding

  • Gründel, Marlene: Supported by the Austrian Science Fund (FWF, projects 10.55776/Y1329 and 10.55776/COE12).
  • Morawietz, Nils: Partially supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).

References

  1. Faisal N. Abu-Khzam, Michael A. Langston, Pushkar Shanbhag, and Christopher T. Symons. Scalable parallel algorithms for FPT problems. Algorithmica, 45(3):269-284, 2006. URL: https://doi.org/10.1007/S00453-006-1214-1.
  2. Marcelo Arenas, Leopoldo Bertossi, and Jan Chomicki. Consistent query answers in inconsistent databases. In Proceedings of the Eighteenth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS '99, pages 68-79, New York, NY, USA, 1999. Association for Computing Machinery. URL: https://doi.org/10.1145/303976.303983.
  3. Max Bannach. Parallel Parameterized Algorithms. PhD thesis, University of Lübeck, Germany, 2020. Google Scholar
  4. Max Bannach, Florian Chudigiewitsch, Kim-Manuel Klein, and Marcel Wienöbst. PACE solver description: Uzl exact solver for one-sided crossing minimization. In Édouard Bonnet and Pawel Rzazewski, editors, 19th International Symposium on Parameterized and Exact Computation, IPEC 2024, volume 321 of LIPIcs, pages 28:1-28:4. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.IPEC.2024.28.
  5. Max Bannach, Florian Chudigiewitsch, and Till Tantau. Existential second-order logic over graphs: Parameterized complexity. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, volume 285 of LIPIcs, pages 3:1-3:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.IPEC.2023.3.
  6. Max Bannach, Florian Chudigiewitsch, and Till Tantau. On the descriptive complexity of vertex deletion problems. In Rastislav Královic and Antonín Kucera, editors, 49th International Symposium on Mathematical Foundations of Computer Science, MFCS 2024, volume 306 of LIPIcs, pages 17:1-17:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.17.
  7. Max Bannach, Malte Skambath, and Till Tantau. On the parallel parameterized complexity of maxsat variants. Journal of Artificial Intelligence Research, 78, 2023. URL: https://doi.org/10.1613/JAIR.1.14748.
  8. Max Bannach, Christoph Stockhusen, and Till Tantau. Fast parallel fixed-parameter algorithms via color coding. In Thore Husfeldt and Iyad A. Kanj, editors, 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, volume 43 of LIPIcs, pages 224-235. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.IPEC.2015.224.
  9. Max Bannach and Till Tantau. Parallel multivariate meta-theorems. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, volume 63 of LIPIcs, pages 4:1-4:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.4.
  10. Max Bannach and Till Tantau. Computing kernels in parallel: Lower and upper bounds. In 13th International Symposium on Parameterized and Exact Computation, IPEC 2018, pages 13:1-13:14, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2018.13.
  11. Max Bannach and Till Tantau. Computing hitting set kernels by AC^0-circuits. Theory of Computing Systems, 64(3):374-399, 2020. URL: https://doi.org/10.1007/s00224-019-09941-z.
  12. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56(1-3):89-113, 2004. URL: https://doi.org/10.1023/B:MACH.0000033116.57574.95.
  13. Peter W. Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, and Koray Kavukcuoglu. Interaction networks for learning about objects, relations and physics. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, editors, Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, pages 4502-4510, 2016. URL: https://proceedings.neurips.cc/paper/2016/hash/3147da8ab4a0437c15ef51a5cc7f2dc4-Abstract.html.
  14. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Journal of Computational Geometry, 4:235-282, 1994. URL: https://doi.org/10.1016/0925-7721(94)00014-X.
  15. Amir Ben-Dor, Ron Shamir, and Zohar Yakhini. Clustering gene expression patterns. Jornal of Computational Biology, 6(3/4):281-297, 1999. URL: https://doi.org/10.1089/106652799318274.
  16. Seymour Benzer. On the topology of the genetic fine structure. Proceedings of the National Academy of Sciences of the United States of America, 45(11):1607-1620, 1959. Google Scholar
  17. Manuel Bodirsky and Santiago Guzmán-Pro. Hereditary first-order logic: the tractable quantifier prefix classes, 2025. URL: https://arxiv.org/abs/2411.10860.
  18. Manuel Bodirsky and Santiago Guzmán Pro. On the computational power of extensional eso, 2025. URL: https://doi.org/10.48550/arXiv.2511.08515.
  19. Egon Börger, Erich Grädel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997. Google Scholar
  20. Daniel Brügmann, Christian Komusiewicz, and Hannes Moser. On generating triangle-free graphs. Electronic Notes in Discrete Mathematics, 32:51-58, 2009. URL: https://doi.org/10.1016/J.ENDM.2009.02.008.
  21. Pablo Burzyn, Flavia Bonomo, and Guillermo Durán. NP-completeness results for edge modification problems. Discrete Applied Mathematics, 154(13):1824-1844, 2006. URL: https://doi.org/10.1016/J.DAM.2006.03.031.
  22. Andrea Calì, Domenico Lembo, and Riccardo Rosati. On the decidability and complexity of query answering over inconsistent and incomplete databases. In Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS '03, pages 260-271, New York, NY, USA, 2003. Association for Computing Machinery. URL: https://doi.org/10.1145/773153.773179.
  23. Marco Cesati and Miriam Di Ianni. Parameterized parallel complexity. In David J. Pritchard and Jeff Reeve, editors, Euro-Par '98 Parallel Processing, 4th International Euro-Par Conference, Southampton, UK, September 1-4, 1998, Proceedings, volume 1470 of Lecture Notes in Computer Science, pages 892-896. Springer, 1998. URL: https://doi.org/10.1007/BFB0057945.
  24. James Cheetham, Frank K. H. A. Dehne, Andrew Rau-Chaplin, Ulrike Stege, and Peter J. Taillon. Solving large FPT problems on coarse-grained parallel machines. Journal of Computing and System Sciences, 67(4):691-706, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00075-8.
  25. Yijia Chen and Jörg Flum. Some lower bounds in parameterized ac^0. Information and Computation, 267:116-134, 2019. URL: https://doi.org/10.1016/J.IC.2019.03.008.
  26. Hyeong-Ah Choi, Kazuo Nakajima, and Chong S. Rim. Graph bipartization and via minimization. SIAM Journal on Discrete Mathematics, 2(1):38-47, 1989. URL: https://doi.org/10.1137/0402004.
  27. Jan Chomicki and Jerzy Marcinkowski. Minimal-change integrity maintenance using tuple deletions. Information and Computation, 197(1):90-121, 2005. URL: https://doi.org/10.1016/j.ic.2004.04.007.
  28. Florian Chudigiewitsch, Marlene Gründel, Christian Komusiewicz, Nils Morawietz, and Till Tantau. The descriptive complexity of relation modification problems, 2026. URL: https://doi.org/10.48550/arXiv.2603.22043.
  29. Christophe Crespelle, Pål Grønås Drange, Fedor V. Fomin, and Petr Golovach. A survey of parameterized algorithms and the complexity of edge modification. Computer Science Review, 48:100556, 2023. URL: https://doi.org/10.1016/j.cosrev.2023.100556.
  30. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  31. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  32. Heinz-Dieter Ebbinghaus and Jörg Flum. Finite Model Theory. Springer, 2nd edition, 2005. URL: https://doi.org/10.1007/3-540-28788-4.
  33. Thomas Eiter, Yuri Gurevich, and Georg Gottlob. Existential second-order logic over strings. Journal of the ACM, 47(1):77-131, 2000. URL: https://doi.org/10.1145/331605.331609.
  34. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica, 71(3):661-701, 2015. URL: https://doi.org/10.1007/S00453-014-9944-Y.
  35. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  36. Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. On the parameterized complexity of graph modification to first-order logic properties. Theory of Computing Systems, 64(2):251-271, 2020. URL: https://doi.org/10.1007/s00224-019-09938-8.
  37. Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Parameterized complexity of elimination distance to first-order logic properties. In ACM Transactions on Computational Logic, volume 23(3), pages 17:1-17:35, 2022. URL: https://doi.org/10.1145/3517129.
  38. Fedor V. Fomin, Saket Saurabh, and Neeldhara Misra. Graph modification problems: A modern perspective. In Jianxin Wang and Chee-Keng Yap, editors, Frontiers in Algorithmics - 9th International Workshop, FAW 2015, volume 9130 of Lecture Notes in Computer Science, pages 3-6. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-19647-3_1.
  39. Georges Gardarin and Stefano Spaccapietra. Integrity of data bases: A general lockout algorithm with deadlock avoidance. In G. M. Nijssen, editor, Modelling in Data Base Management Systems, Proceeding of the IFIP Working Conference on Modelling in Data Base Management Systems, Freudenstadt, Germany, pages 395-412. North-Holland, 1976. Google Scholar
  40. Georg Gottlob, Phokion G. Kolaitis, and Thomas Schwentick. Existential second-order logic over graphs: Charting the tractability frontier. Journal of the ACM, 51(2):312-362, 2004. URL: https://doi.org/10.1145/972639.972646.
  41. Erich Grädel, Phokion G. Kolaitis, Leonid Libkin, Maarten Marx, Joel Spencer, Moshe Y. Vardi, Yde Venema, and Scott Weinstein. Finite Model Theory and Its Applications. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2007. URL: https://doi.org/10.1007/3-540-68804-8.
  42. Martin Grohe. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory, volume 47 of Lecture Notes in Logic. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781139028868.
  43. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, volume 3162 of Lecture Notes in Computer Science, pages 162-173. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_15.
  44. Mohsen MD Hassan and Gary L Hogg. A review of graph theory application to the facilities layout problem. Omega, 15(4):291-300, 1987. URL: https://doi.org/10.1016/0305-0483(87)90017-X.
  45. Steef Hegeman, Jan Martens, and Alfons Laarman. Uniformity within parameterized circuit classes. In Akanksha Agrawal and Erik Jan van Leeuwen, editors, 20th International Symposium on Parameterized and Exact Computation, IPEC 2025, volume 358 of LIPIcs, pages 27:1-27:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.IPEC.2025.27.
  46. Falk Hüffner, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. Fixed-parameter algorithms for cluster vertex deletion. Theory of Computing Systems, 47(1):196-217, 2010. URL: https://doi.org/10.1007/S00224-008-9150-X.
  47. Neil Immerman. Descriptive Complexity. Springer Verlag, 1998. Google Scholar
  48. Michael Jünger and Petra Mutzel. Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica, 16(1):33-59, 1996. URL: https://doi.org/10.1007/BF02086607.
  49. Athanasios L. Konstantinidis and Charis Papadopoulos. Cluster deletion on interval graphs and split related graphs. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, volume 138 of LIPIcs, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.12.
  50. Paraschos Koutris, Xiating Ouyang, and Jef Wijsen. Consistent query answering for primary keys on path queries. In Proceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS'21, pages 215-232, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3452021.3458334.
  51. Robert Kudelic. Feedback Arc Set - A History of the Problem and Algorithms. Springer Briefs in Computer Science. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-10515-9.
  52. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  53. Kun Liu and Evimaria Terzi. Towards identity anonymization on graphs. In Proceedings of the 2008 ACM SIGMOD International Conference on Management of Data, SIGMOD '08, pages 93-106, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1376616.1376629.
  54. Andrei Lopatenko and Leopoldo Bertossi. Complexity of consistent query answering in databases under cardinality-based and incremental repair semantics. In Thomas Schwentick and Dan Suciu, editors, Database Theory - ICDT 2007, pages 179-193, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  55. Xuesong Lu, Yi Song, and Stéphane Bressan. Fast identity anonymization on graphs. In Stephen W. Liddle, Klaus-Dieter Schewe, A Min Tjoa, and Xiaofang Zhou, editors, Database and Expert Systems Applications - 23rd International Conference, DEXA 2012, volume 7446 of Lecture Notes in Computer Science, pages 281-295. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32600-4_21.
  56. Frederico Mancini. Graph modification problems related to graph classes. PhD thesis, University of Bergen, Norway, May 2008. Google Scholar
  57. Laure Morelle, Ignasi Sau, and Dimitrios M. Thilikos. Graph modification of bounded size to minor-closed classes as fast as vertex deletion. In Anne Benoit, Haim Kaplan, Sebastian Wild, and Grzegorz Herman, editors, 33rd Annual European Symposium on Algorithms, ESA 2025, volume 351 of LIPIcs, pages 7:1-7:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.7.
  58. Assaf Natanzon, Ron Shamir, and Roded Sharan. Complexity classification of some edge modification problems. Discrete Applied Mathematics, 113(1):109-128, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00391-7.
  59. Romeo Rizzi, Vineet Bafna, Sorin Istrail, and Giuseppe Lancia. Practical algorithms and fixed-parameter tractability for the single individual snp haplotyping problem. In Proceedings of the Second International Workshop on Algorithms in Bioinformatics, WABI '02, pages 29-43, Berlin, Heidelberg, 2002. Springer-Verlag. URL: https://doi.org/10.1007/3-540-45784-4_3.
  60. Zimo Sheng and Mingyu Xiao. A discharging method: Improved kernels for edge triangle packing and covering. In Weili Wu and Guangmo Tong, editors, Computing and Combinatorics - 29th International Conference, COCOON 2023, volume 14423 of Lecture Notes in Computer Science, pages 171-183. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49193-1_13.
  61. Sławomir Staworko and Jan Chomicki. Consistent query answers in the presence of universal constraints. Information Systems, 35(1):1-22, 2010. URL: https://doi.org/10.1016/j.is.2009.03.004.
  62. Till Tantau. Existential second-order logic over graphs: A complete complexity-theoretic classification. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, pages 703-715, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.703.
  63. Shaobin Xu, Minghui Sun, and Jun Qin. A high-scalability graph modification system for large-scale networks. Computer Graphics Forum, 43(6), 2024. URL: https://doi.org/10.1111/CGF.15191.
  64. Mihalis Yannakakis. Node-and edge-deletion NP-complete problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC '78, pages 253-264, New York, NY, USA, 1978. Association for Computing Machinery. URL: https://doi.org/10.1145/800133.804355.
  65. Mihalis Yannakakis. Edge-deletion problems. SIAM Journal on Computing, 10(2):297-309, 1981. URL: https://doi.org/10.1137/0210021.
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