Document Open Access Logo

A Fine-Grained Analogue of Schaefer’s Theorem in P: Dichotomy of Exists^k-Forall-Quantified First-Order Graph Properties

Authors Karl Bringmann, Nick Fischer, Marvin Künnemann

PDF
Thumbnail PDF

File

LIPIcs.CCC.2019.31.pdf
  • Filesize: 0.67 MB
  • 27 pages

Document Identifiers

Author Details

Karl Bringmann
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany
Nick Fischer
  • Max Planck Institute for Informatics, Saarbrücken Graduate School of Computer Science, Saarbrücken, Germany
Marvin Künnemann
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany

Cite As Get BibTex

Karl Bringmann, Nick Fischer, and Marvin Künnemann. A Fine-Grained Analogue of Schaefer’s Theorem in P: Dichotomy of Exists^k-Forall-Quantified First-Order Graph Properties. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 31:1-31:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.31

Abstract

An important class of problems in logics and database theory is given by fixing a first-order property psi over a relational structure, and considering the model-checking problem for psi. Recently, Gao, Impagliazzo, Kolokolova, and Williams (SODA 2017) identified this class as fundamental for the theory of fine-grained complexity in P, by showing that the (Sparse) Orthogonal Vectors problem is complete for this class under fine-grained reductions. This raises the question whether fine-grained complexity can yield a precise understanding of all first-order model-checking problems. Specifically, can we determine, for any fixed first-order property psi, the exponent of the optimal running time O(m^{c_psi}), where m denotes the number of tuples in the relational structure?
Towards answering this question, in this work we give a dichotomy for the class of exists^k-forall-quantified graph properties. For every such property psi, we either give a polynomial-time improvement over the baseline O(m^k)-time algorithm or show that it requires time m^{k-o(1)} under the hypothesis that MAX-3-SAT has no O((2-epsilon)^n)-time algorithm. More precisely, we define a hardness parameter h = H(psi) such that psi can be decided in time O(m^{k-epsilon}) if h <=2 and requires time m^{k-o(1)} for h >= 3 unless the h-uniform HyperClique hypothesis fails. This unveils a natural hardness hierarchy within first-order properties: for any h >= 3, we show that there exists a exists^k-forall-quantified graph property psi with hardness H(psi)=h that is solvable in time O(m^{k-epsilon}) if and only if the h-uniform HyperClique hypothesis fails. Finally, we give more precise upper and lower bounds for an exemplary class of formulas with k=3 and extend our classification to a counting dichotomy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Fine-grained Complexity
  • Hardness in P
  • Hyperclique Conjecture
  • Constrained Triangle Detection

Metrics

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

References

  1. Amir Abboud, Arturs Backurs, Karl Bringmann, and Marvin Künnemann. Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-and-Solve. In Proceedings of the 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS '17, pages 192-203, 2017. URL: https://doi.org/10.1109/FOCS.2017.26.
  2. Amir Abboud, Arturs Backurs, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Or Zamir. Subtree Isomorphism Revisited. ACM Trans. Algorithms, 14(3):27:1-27:23, 2018. URL: https://doi.org/10.1145/3093239.
  3. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the Current Clique Algorithms Are Optimal, So is Valiant’s Parser. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS '15, pages 98-117, 2015. URL: https://doi.org/10.1109/FOCS.2015.16.
  4. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight Hardness Results for LCS and Other Sequence Similarity Measures. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS '15, pages 59-78, 2015. URL: https://doi.org/10.1109/FOCS.2015.14.
  5. Amir Abboud, Karl Bringmann, Holger Dell, and Jesper Nederlof. More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 253-266. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188938.
  6. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 377-391, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch28.
  7. Amir Abboud, Ryan Williams, and Huacheng Yu. More Applications of the Polynomial Method to Algorithm Design. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 218-230, 2015. URL: http://dl.acm.org/citation.cfm?id=2722129.2722146.
  8. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of Faster Alignment of Sequences. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming, pages 39-51. Springer Berlin Heidelberg, 2014. Google Scholar
  9. Serge Abiteboul, Richard Hull, and Victor Vianu, editors. Foundations of Databases: The Logical Level. Addison-Wesley Longman Publishing Co., Inc., 1st edition, 1995. Google Scholar
  10. Alfred V. Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalences Among Relational Expressions. SIAM J. Comput., 8(2):218-246, 1979. URL: https://doi.org/10.1137/0208017.
  11. Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial Representations of Threshold Functions and Algorithmic Applications. In Proceedings of the 2016 IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS '16, pages 467-476, 2016. URL: https://doi.org/10.1109/FOCS.2016.57.
  12. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and Counting Given Length Cycles. Algorithmica, 17(3):209-223, 1997. URL: https://doi.org/10.1007/BF02523189.
  13. Arturs Backurs and Piotr Indyk. Edit Distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proceedings of the 47th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2015, pages 51-58. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746612.
  14. Arturs Backurs and Piotr Indyk. Which Regular Expression Patterns Are Hard to Match? In Proceedings of the 2016 IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS '16, pages 457-466, 2016. URL: https://doi.org/10.1109/FOCS.2016.56.
  15. Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Towards tight approximation bounds for graph diameter and eccentricities. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 267-280, 2018. URL: https://doi.org/10.1145/3188745.3188950.
  16. Guillaume Bagan, Arnaud Durand, and Etienne Grandjean. On Acyclic Conjunctive Queries and Constant Delay Enumeration. In Jacques Duparc and Thomas A. Henzinger, editors, Computer Science Logic, pages 208-222. Springer Berlin Heidelberg, 2007. Google Scholar
  17. Christoph Berkholz, Jens Keppeler, and Nicole Schweikardt. Answering Conjunctive Queries Under Updates. In Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS '17, pages 303-318. ACM, 2017. URL: https://doi.org/10.1145/3034786.3034789.
  18. Karl Bringmann. Why Walking the Dog Takes Time: Fréchet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails. In Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, FOCS '14, pages 661-670, 2014. URL: https://doi.org/10.1109/FOCS.2014.76.
  19. Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A Dichotomy for Regular Expression Membership Testing. In Proceedings of the 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS '17, pages 307-318, 2017. URL: https://doi.org/10.1109/FOCS.2017.36.
  20. Karl Bringmann and Marvin Künnemann. Quadratic Conditional Lower Bounds for String Problems and Dynamic Time Warping. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS '15, pages 79-97, 2015. URL: https://doi.org/10.1109/FOCS.2015.15.
  21. Karl Bringmann and Marvin Künnemann. Multivariate Fine-Grained Complexity of Longest Common Subsequence. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1216-1235, 2018. URL: https://doi.org/10.1137/1.9781611975031.79.
  22. Andrei A. Bulatov. A Dichotomy Theorem for Constraints on a Three-Element Set. In Proceedings of the 2002 IEEE 43rd Annual Symposium on Foundations of Computer Science, FOCS '02, pages 649-658, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181990.
  23. Andrei A. Bulatov. Tractable conservative Constraint Satisfaction Problems. In 18th IEEE Symposium on Logic in Computer Science, page 321, 2003. URL: https://doi.org/10.1109/LICS.2003.1210072.
  24. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In Proceedings of the 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS '17, pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  25. Andrei A. Bulatov. Constraint satisfaction problems: complexity and algorithms. SIGLOG News, 5(4):4-24, 2018. URL: https://doi.org/10.1145/3292048.3292050.
  26. Andrei A. Bulatov, Peter Jeavons, and Andrei A. Krokhin. Classifying the Complexity of Constraints Using Finite Algebras. SIAM J. Comput., 34(3):720-742, 2005. URL: https://doi.org/10.1137/S0097539700376676.
  27. Nofar Carmeli and Markus Kröll. Enumeration Complexity of Conjunctive Queries with Functional Dependencies. In ICDT, volume 98 of LIPIcs, pages 11:1-11:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  28. Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic Extensions of the Strong Exponential Time Hypothesis and Consequences for Non-reducibility. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 261-270, 2016. URL: https://doi.org/10.1145/2840728.2840746.
  29. Timothy M. Chan and Ryan Williams. Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1246-1255, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884522.
  30. Ashok K. Chandra and David Harel. Structure and Complexity of Relational Queries. J. Comput. Syst. Sci., 25(1):99-128, 1982. URL: https://doi.org/10.1016/0022-0000(82)90012-5.
  31. Ashok K. Chandra and Philip M. Merlin. Optimal Implementation of Conjunctive Queries in Relational Data Bases. In Proceedings of the 9th Annual ACM SIGACT Symposium on Theory of Computing, STOC 1977, pages 77-90, 1977. URL: https://doi.org/10.1145/800105.803397.
  32. Hubie Chen. A rendezvous of logic, complexity, and algebra. SIGACT News, 37(4):85-114, 2006. URL: https://doi.org/10.1145/1189056.1189076.
  33. Nadia Creignou and Miki Hermann. Complexity of Generalized Satisfiability Counting Problems. Inf. Comput., 125(1):1-12, 1996. URL: https://doi.org/10.1006/inco.1996.0016.
  34. Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity classifications of Boolean Constraint Satisfaction problems. SIAM, 2001. URL: https://doi.org/10.1137/1.9780898718546.
  35. Víctor Dalmau. Some dichotomy theorems on constant free Boolean formulas. Technical Report LSI-97-43-R, Departament LSI, Universitat Politècnica de Catalunya, 1997. Google Scholar
  36. Holger Dell, Marc Roth, and Philip Wellnitz. Counting Answers to Existential Questions. arXiv e-prints, February 2019. URL: http://arxiv.org/abs/1902.04960.
  37. Rodney G. Downey, Michael R. Fellows, and Udayan Taylor. The Parameterized Complexity of Relational Database Queries and an Improved Characterization of W[1]. In First Conference of the Centre for Discrete Mathematics and Theoretical Computer Science, pages 194-213, 1996. Google Scholar
  38. Tomás Feder and Moshe Y. Vardi. Monotone monadic SNP and constraint satisfaction. In Proceedings of the 25th Annual ACM SIGACT Symposium on Theory of Computing, STOC 1993, pages 612-622, 1993. URL: https://doi.org/10.1145/167088.167245.
  39. Tomás Feder and Moshe Y. Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM J. Comput., 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  40. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  41. Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and Ryan Williams. Completeness for First-order Properties on Sparse Structures with Algorithmic Applications. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 2162-2181, 2017. URL: http://dl.acm.org/citation.cfm?id=3039686.3039827.
  42. Michel Habib, Christophe Paul, and Laurent Viennot. A Synthesis on Partition Refinement: A Useful Routine for Strings, Graphs, Boolean Matrices and Automata. In STACS 98, 15th Annual Symposium on Theoretical Aspects of Computer Science, pages 25-38, 1998. URL: https://doi.org/10.1007/BFb0028546.
  43. Pavol Hell and Jaroslav Nesetril. On the complexity of H-coloring. J. Comb. Theory, Ser. B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
  44. Neil Immerman. Upper and Lower Bounds for First Order Expressibility. J. Comput. Syst. Sci., 25(1):76-98, 1982. URL: https://doi.org/10.1016/0022-0000(82)90011-3.
  45. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  46. Peter Jeavons. On the Algebraic Structure of Combinatorial Problems. Theor. Comput. Sci., 200(1-2):185-204, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00230-2.
  47. Peter Jeavons, David A. Cohen, and Marc Gyssens. Closure properties of constraints. J. ACM, 44(4):527-548, 1997. URL: https://doi.org/10.1145/263867.263489.
  48. Robert Krauthgamer and Ohad Trabelsi. Conditional Lower Bounds for All-Pairs Max-Flow. ACM Trans. Algorithms, 14(4):42:1-42:15, 2018. URL: https://doi.org/10.1145/3212510.
  49. Andrea Lincoln, Virginia Vassilevska Williams, and Ryan Williams. Tight Hardness for Shortest Cycles and Paths in Sparse Graphs. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1236-1252, 2018. URL: http://dl.acm.org/citation.cfm?id=3174304.3175350.
  50. Christos H. Papadimitriou and Mihalis Yannakakis. On the Complexity of Database Queries. J. Comput. Syst. Sci., 58(3):407-427, 1999. URL: https://doi.org/10.1006/jcss.1999.1626.
  51. Mihai Patrascu and Ryan Williams. On the Possibility of Faster SAT Algorithms. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '10, pages 1065-1075, 2010. URL: https://doi.org/10.1137/1.9781611973075.86.
  52. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the 45th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2013, pages 515-524, 2013. URL: https://doi.org/10.1145/2488608.2488673.
  53. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the 10th Annual ACM SIGACT Symposium on Theory of Computing, STOC 1978, pages 216-226, 1978. URL: https://doi.org/10.1145/800133.804350.
  54. Moshe Y. Vardi. The Complexity of Relational Query Languages (Extended Abstract). In Proceedings of the 14th Annual ACM SIGACT Symposium on Theory of Computing, STOC 1982, pages 137-146, 1982. URL: https://doi.org/10.1145/800070.802186.
  55. Moshe Y. Vardi. On the Complexity of Bounded-Variable Queries. In Proceedings of the 14th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 266-276, 1995. URL: https://doi.org/10.1145/212433.212474.
  56. Virginia Vassilevska and Ryan Williams. Finding, Minimizing, and Counting Weighted Subgraphs. In Proceedings of the 41st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2009, pages 455-464. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536477.
  57. Ryan Williams. A New Algorithm for Optimal 2-constraint Satisfaction and Its Implications. Theor. Comput. Sci., 348(2):357-365, December 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.023.
  58. Ryan Williams. Faster decision of first-order graph properties. In Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 80:1-80:6, 2014. URL: https://doi.org/10.1145/2603088.2603121.
  59. Ryan Williams. On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1207-1215, 2018. URL: https://doi.org/10.1137/1.9781611975031.78.
  60. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In Proceedings of the 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS '17, pages 331-342, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact