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Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Authors Silvia Butti , Víctor Dalmau

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Author Details

Silvia Butti
  • Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
Víctor Dalmau
  • Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain

Funding

  • Butti, Silvia: The project that gave rise to these results received the support of a fellowship from "la Caixa" Foundation (ID 100010434). The fellowship code is LCF/BQ/DI18/11660056. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 713673.
  • Dalmau, Víctor: Víctor Dalmau was supported by MICCIN grant PID2019-109137GB-C22.

Cite AsGet BibTex

Silvia Butti and Víctor Dalmau. Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.27

Abstract

Given a pair of graphs 𝐀 and 𝐁, the problems of deciding whether there exists either a homomorphism or an isomorphism from 𝐀 to 𝐁 have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where 𝐀 and 𝐁 are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Constraint and logic programming
Keywords
  • Weisfeiler-Leman algorithm
  • Sherali-Adams hierarchy
  • Graph homomorphism
  • Constraint Satisfaction Problem

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References

  1. Dana Angluin. Local and global properties in networks of processors (extended abstract). In Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC ’80, page 82–93, New York, NY, USA, 1980. Association for Computing Machinery. URL: https://doi.org/10.1145/800141.804655.
  2. Albert Atserias, Andrei A. Bulatov, and Anuj Dawar. Affine systems of equations and counting infinitary logic. Theor. Comput. Sci., 410(18):1666-1683, 2009. URL: https://doi.org/10.1016/j.tcs.2008.12.049.
  3. Albert Atserias and Elitza Maneva. Sherali-adams relaxations and indistinguishability in counting logics. SIAM Journal on Computing, 42(1):112-137, 2013. URL: https://doi.org/10.1137/120867834.
  4. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, page 684–697, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2897518.2897542.
  5. László Babai, Paul Erdös, and Stanley M. Selkow. Random graph isomorphism. SIAM J. Comput., 9(3):628-635, 1980. URL: https://doi.org/10.1137/0209047.
  6. Libor Barto. The collapse of the bounded width hierarchy. Journal of Logic and Computation, 26(3):923-943, 2016. Google Scholar
  7. Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, and Stanislav Zivný. The power of the combined basic linear programming and affine relaxation for promise constraint satisfaction problems. SIAM J. Comput., 49(6):1232-1248, 2020. URL: https://doi.org/10.1137/20M1312745.
  8. A. A. Bulatov. A dichotomy theorem for nonuniform CSPs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 319-330, October 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  9. Silvia Butti and Victor Dalmau. The complexity of the distributed constraint satisfaction problem. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), volume 187 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.20.
  10. Jin-yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identifications. Comb., 12(4):389-410, 1992. URL: https://doi.org/10.1007/BF01305232.
  11. Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 350-359. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.45.
  12. Víctor Dalmau and Andrei A. Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. ACM Trans. Comput. Theory, 5(4):15:1-15:25, 2013. URL: https://doi.org/10.1145/2540090.
  13. Víctor Dalmau, Andrei A. Krokhin, and Rajsekar Manokaran. Towards a characterization of constant-factor approximable finite-valued CSPs. J. Comput. Syst. Sci., 97:14-27, 2018. URL: https://doi.org/10.1016/j.jcss.2018.03.003.
  14. Holger Dell, Martin Grohe, and Gaurav Rattan. Lovász meets weisfeiler and leman. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 40:1-40:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.40.
  15. 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
  16. Ferdinando Fioretto, Enrico Pontelli, and William Yeoh. Distributed constraint optimization problems and applications: A survey. J. Artif. Int. Res., 61(1):623-698, 2018. Google Scholar
  17. Konstantinos Georgiou, Avner Magen, and Madhur Tulsiani. Optimal sherali-adams gaps from pairwise independence. In Irit Dinur, Klaus Jansen, Joseph Naor, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings, volume 5687 of Lecture Notes in Computer Science, pages 125-139. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_10.
  18. Mrinalkanti Ghosh and Madhur Tulsiani. From weak to strong linear programming gaps for all constraint satisfaction problems. Theory Comput., 14(1):1-33, 2018. URL: https://doi.org/10.4086/toc.2018.v014a010.
  19. Martin Grohe, Kristian Kersting, Martin Mladenov, and Erkal Selman. Dimension reduction via colour refinement. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 505-516. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_42.
  20. Martin Grohe and Martin Otto. Pebble games and linear equations. J. Symb. Log., 80(3):797-844, 2015. URL: https://doi.org/10.1017/jsl.2015.28.
  21. Pavol Hell and Jaroslav Nešetřil. Graphs and Homomorphisms. Oxford University Press, 2004. Google Scholar
  22. Neil Immerman and Eric Lander. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective: In Honor of Juris Hartmanis on the Occasion of His Sixtieth Birthday, pages 59-81, 1990. Google Scholar
  23. Amit Kumar, Rajsekar Manokaran, Madhur Tulsiani, and Nisheeth K. Vishnoi. On lp-based approximability for strict csps. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1560-1573. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.121.
  24. Gabor Kun, Ryan O’Donnell, Suguru Tamaki, Yuichi Yoshida, and Yuan Zhou. Linear programming, width-1 CSPs, and robust satisfaction. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS ’12, page 484–495, New York, NY, USA, 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2090236.2090274.
  25. Benoît Larose, Cynthia Loten, and Claude Tardif. A characterisation of first-order constraint satisfaction problems. Log. Methods Comput. Sci., 3(4), 2007. URL: https://doi.org/10.2168/LMCS-3(4:6)2007.
  26. AA Leman and B Weisfeiler. A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsiya, 2(9):12-16, 1968. Google Scholar
  27. László Lovász and Alexander Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim., 1(2):166-190, 1991. URL: https://doi.org/10.1137/0801013.
  28. Peter Malkin. Sherali–adams relaxations of graph isomorphism polytopes. Discrete Optimization, 12:73-97, 2014. Google Scholar
  29. Jaroslav Nešetřil and Vojtěch Rödl. Chromatically optimal rigid graphs. Journal of Combinatorial Theory, Series B, 46(2):133-141, 1989. URL: https://doi.org/10.1016/0095-8956(89)90039-7.
  30. Motakuri V. Ramana, Edward R. Scheinerman, and Daniel Ullman. Fractional isomorphism of graphs. Discrete Mathematics, 132(1):247-265, 1994. URL: https://doi.org/10.1016/0012-365X(94)90241-0.
  31. Edward R Scheinerman and Daniel H Ullman. Fractional graph theory: a rational approach to the theory of graphs. Courier Corporation, 2011. Google Scholar
  32. Hanif D. Sherali and Warren P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math., 3(3):411-430, 1990. URL: https://doi.org/10.1137/0403036.
  33. Johan Thapper and Stanislav Zivný. The power of sherali-adams relaxations for general-valued csps. SIAM J. Comput., 46(4):1241-1279, 2017. URL: https://doi.org/10.1137/16M1079245.
  34. Gottfried Tinhofer. Graph isomorphism and theorems of birkhoff type. Computing, 36(4):285-300, 1986. URL: https://doi.org/10.1007/BF02240204.
  35. Gottfried Tinhofer. A note on compact graphs. Discret. Appl. Math., 30(2-3):253-264, 1991. URL: https://doi.org/10.1016/0166-218X(91)90049-3.
  36. Yuichi Yoshida and Yuan Zhou. Approximation schemes via sherali-adams hierarchy for dense constraint satisfaction problems and assignment problems. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 423-438. ACM, 2014. URL: https://doi.org/10.1145/2554797.2554836.
  37. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 331-342, October 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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