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Parameterized Complexity Classification for Interval Constraints

Authors Konrad K. Dabrowski , Peter Jonsson , Sebastian Ordyniak , George Osipov , Marcin Pilipczuk , Roohani Sharma

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Author Details

Konrad K. Dabrowski
  • School of Computing, Newcastle University, UK
Peter Jonsson
  • Department of Computer and Information Science, Linköping University, Sweden
Sebastian Ordyniak
  • School of Computing, University of Leeds, UK
George Osipov
  • Department of Computer and Information Science, Linköping University, Sweden
Marcin Pilipczuk
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
  • IT University Copenhagen, Denmark
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Jonsson, Peter: Partially supported by the Swedish Research Council (VR) under grant 2021-0437 and the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
  • Ordyniak, Sebastian: Supported by the Engineering and Physical Sciences Research Council (EPSRC, project EP/V00252X/1).
  • Osipov, George: Supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
  • Pilipczuk, Marcin: During this research Marcin was part of BARC, supported by the VILLUM Foundation grant 16582.

Cite AsGet BibTex

Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov, Marcin Pilipczuk, and Roohani Sharma. Parameterized Complexity Classification for Interval Constraints. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.11

Abstract

Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most k constraints, where k is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen’s interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set A of 13 basic comparison relations such as "precedes" or "during" for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP(Γ) for all Γ ⊆ A. IA is sometimes extended with unions of the relations in A or first-order definable relations over A, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP(A) in general, we then consider (parameterized) approximation algorithms. We first show that MinCSP(A) cannot be polynomial-time approximated within any constant factor and continue by presenting a factor-2 fpt-approximation algorithm. Once again, this algorithm has its roots in flow augmentation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • (minimum) constraint satisfaction problem
  • Allen’s interval algebra
  • parameterized complexity
  • cut problems

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References

  1. Akanksha Agrawal, Daniel Lokshtanov, Amer E. Mouawad, and Saket Saurabh. Simultaneous feedback vertex set: A parameterized perspective. ACM Transactions on Computation Theory, 10(4):1-25, 2018. Google Scholar
  2. Akanksha Agrawal, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Simultaneous feedback edge set: a parameterized perspective. Algorithmica, 83(2):753-774, 2021. Google Scholar
  3. James F. Allen. Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11):832-843, 1983. Google Scholar
  4. James F. Allen and Johannes A. G. M. Koomen. Planning using a temporal world model. In Proc. 8th International Joint Conference on Artificial Intelligence (IJCAI-1983), pages 741-747, 1983. Google Scholar
  5. Kristóf Bérczi, Alexander Göke, Lydia Mirabel Mendoza Cadena, and Matthias Mnich. Resolving infeasibility of linear systems: A parameterized approach. CoRR, abs/2209.02017, 2022. Google Scholar
  6. Benjamin Bergougnoux, Eduard Eiben, Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Towards a polynomial kernel for directed feedback vertex set. Algorithmica, 83(5):1201-1221, 2021. Google Scholar
  7. Leopoldo Bertossi and Jan Chomicki. Query answering in inconsistent databases. In Logics for Emerging Applications of Databases, pages 43-83. Springer, 2004. Google Scholar
  8. Manuel Bodirsky. Complexity of Infinite-Domain Constraint Satisfaction. Cambridge University Press, 2021. Google Scholar
  9. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Proc. 35th International Colloquium on Automata, Languages and Programming (ICALP-2008), pages 184-196, 2008. Google Scholar
  10. Manuel Bodirsky, Peter Jonsson, Barnaby Martin, Antoine Mottet, and Zaneta Semanisinová. Complexity classification transfer for CSPs via algebraic products. CoRR, abs/2211.03340, 2022. Google Scholar
  11. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2):9:1-9:41, 2010. Google Scholar
  12. Marthe Bonamy, Łukasz Kowalik, Jesper Nederlof, Michał Pilipczuk, Arkadiusz Socała, and Marcin Wrochna. On directed feedback vertex set parameterized by treewidth. In Proc. 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG-2018), volume 11159, pages 65-78, 2018. Google Scholar
  13. Édouard Bonnet, László Egri, and Dániel Marx. Fixed-parameter approximability of Boolean MinCSPs. In Proc. 24th Annual European Symposium on Algorithms (ESA-2016), pages 18:1-18:18, 2016. Google Scholar
  14. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In Proc. 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS-2017), pages 319-330, 2017. Google Scholar
  15. Andrei A. Bulatov, Peter Jeavons, and Andrei A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, 2005. Google Scholar
  16. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. In Proc. 40th Annual ACM Symposium on Theory of Computing (STOC-2008), pages 177-186, 2008. Google Scholar
  17. Rajesh Chitnis, Marek Cygan, Mohammataghi Hajiaghayi, and Dániel Marx. Directed subset feedback vertex set is fixed-parameter tractable. ACM Transactions on Algorithms, 11(4):1-28, 2015. Google Scholar
  18. Jan Chomicki and Jerzy Marcinkowski. Minimal-change integrity maintenance using tuple deletions. Information and Computation, 197(1-2):90-121, 2005. Google Scholar
  19. Jean-François Condotta, Issam Nouaouri, and Michael Sioutis. A SAT approach for maximizing satisfiability in qualitative spatial and temporal constraint networks. In Proc. 15th International Conference on the Principles of Knowledge Representation and Reasoning (KR-2016), 2016. Google Scholar
  20. Robert Crowston, Gregory Gutin, Mark Jones, and Anders Yeo. Parameterized complexity of satisfying almost all linear equations over 𝔽₂. Theory of Computing Systems, 52(4):719-728, 2013. Google Scholar
  21. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. Google Scholar
  22. Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, and George Osipov. Resolving inconsistencies in simple temporal problems: A parameterized approach. In Proc. 36th AAAI Conference on Artificial Intelligence, (AAAI-2022), pages 3724-3732, 2022. Google Scholar
  23. Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov, Marcin Pilipczuk, and Roohani Sharma. Parameterized complexity classification for interval constraints. CoRR, abs/2305.13889, 2023. Google Scholar
  24. Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov, and Magnus Wahlström. Almost consistent systems of linear equations. In Proc. 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2023), pages 3179-3217, 2023. Google Scholar
  25. Rina Dechter, Itay Meiri, and Judea Pearl. Temporal constraint networks. Artificial Intelligence, 49(1-3):61-95, 1991. Google Scholar
  26. Pascal Denis and Philippe Muller. Predicting globally-coherent temporal structures from texts via endpoint inference and graph decomposition. In Proc. 22nd International Joint Conference on Artificial Intelligence (IJCAI-2011), pages 1788-1793, 2011. Google Scholar
  27. Eduard Eiben, Clément Rambaud, and Magnus Wahlström. On the parameterized complexity of symmetric directed multicut. In Proc. 17th International Symposium on Parameterized and Exact Computation (IPEC-2022), volume 249, pages 11:1-11:17, 2022. Google Scholar
  28. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998. Google Scholar
  29. Alexander Göke, Dániel Marx, and Matthias Mnich. Hitting long directed cycles is fixed-parameter tractable. In Proc. 47th International Colloquium on Automata, Languages, and Programming (ICALP-2020), volume 168, pages 59:1-59:18, 2020. Google Scholar
  30. Alexander Göke, Dániel Marx, and Matthias Mnich. Parameterized algorithms for generalizations of directed feedback vertex set. Discrete Optimization, 46:100740, 2022. Google Scholar
  31. Martin Charles Golumbic and Ron Shamir. Complexity and algorithms for reasoning about time: A graph-theoretic approach. Journal of the ACM, 40(5):1108-1133, 1993. Google Scholar
  32. Venkatesan Guruswami, Rajsekar Manokaran, and Prasad Raghavendra. Beating the random ordering is hard: inapproximability of maximum acyclic subgraph. In Proc. 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS-2008), pages 573-582, 2008. Google Scholar
  33. Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David P. Williamson. The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863-1920, 2000. Google Scholar
  34. Subhash Khot. On the power of unique 2-prover 1-round games. In Proc. 24th Annual ACM Symposium on Theory of Computing (STOC-2002), pages 767-775, 2002. Google Scholar
  35. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Solving hard cut problems via flow-augmentation. In Proc. 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA-2021), pages 149-168, 2021. Google Scholar
  36. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Directed flow-augmentation. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC-2022), pages 938-947, 2022. Google Scholar
  37. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Flow-augmentation III: complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints. In Proc. 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2023), pages 3218-3228, 2023. Google Scholar
  38. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Flow-augmentation I: directed graphs. CoRR, abs/2111.03450, 2023. Google Scholar
  39. Eun Jung Kim, Tomáš Masařík, Marcin Pilipczuk, Roohani Sharma, and Magnus Wahlström. On weighted graph separation problems and flow-augmentation. CoRR, abs/2208.14841, 2022. Google Scholar
  40. Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. The complexity of general-valued CSPs. SIAM Journal on Computing, 46(3):1087-1110, 2017. Google Scholar
  41. Andrei A. Krokhin, Peter Jeavons, and Peter Jonsson. Reasoning about temporal relations: The tractable subalgebras of Allen’s interval algebra. Journal of the ACM, 50(5):591-640, 2003. Google Scholar
  42. Andrei A. Krokhin and Jakub Oprsal. An invitation to the promise constraint satisfaction problem. ACM SIGLOG News, 9(3):30-59, 2022. Google Scholar
  43. Victor Lagerkvist. A new characterization of restriction-closed hyperclones. In Proc. 50th IEEE International Symposium on Multiple-Valued Logic (ISMVL-2020), pages 303-308, 2020. Google Scholar
  44. Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Fpt-approximation for fpt problems. In Proc. 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA-2021), pages 199-218, 2021. Google Scholar
  45. Daniel Lokshtanov, M. S. Ramanujan, and Saket Saurabh. Linear time parameterized algorithms for subset feedback vertex set. ACM Transactions on Algorithms, 14(1):7:1-7:37, 2018. Google Scholar
  46. Daniel Lokshtanov, M. S. Ramanujan, and Saket Saurabh. When recursion is better than iteration: A linear-time algorithm for acyclicity with few error vertices. In Proc. 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2018), pages 1916-1933, 2018. Google Scholar
  47. Dániel Marx and Igor Razgon. Constant ratio fixed-parameter approximation of the edge multicut problem. Information Processing Letters, 109(20):1161-1166, 2009. Google Scholar
  48. Lenka Mudrová and Nick Hawes. Task scheduling for mobile robots using interval algebra. In Proc. 2015 IEEE International Conference on Robotics and Automation (ICRA-2015), pages 383-388, 2015. Google Scholar
  49. Richard N. Pelavin and James F. Allen. A model for concurrent actions having temporal extent. In Proc. 6th National Conference on Artificial Intelligence (AAAI-1987), pages 246-250, 1987. Google Scholar
  50. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proc. 40th Annual ACM Symposium on Theory of Computing (STOC-2008), pages 245-254, 2008. Google Scholar
  51. Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. Journal of Computer and System Sciences, 75(8):435-450, 2009. Google Scholar
  52. Fei Song and Robin Cohen. The interpretation of temporal relations in narrative. In Proc. 7th National Conference on Artificial Intelligence (AAAI-1988), pages 745-750, 1988. Google Scholar
  53. Marc B. Vilain and Henry A. Kautz. Constraint propagation algorithms for temporal reasoning. In Proc. 5th National Conference on Artificial Intelligence (AAAI-1986), pages 377-382, 1986. Google Scholar
  54. Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. Journal of the ACM, 67(5):30:1-30:78, 2020. Google Scholar
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