Document Open Access Logo

Dismantlability, Connectedness, and Mixing in Relational Structures

Authors Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, Benoît Larose

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.29.pdf
  • Filesize: 0.5 MB
  • 15 pages

Document Identifiers

Author Details

Raimundo Briceño
  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Andrei A. Bulatov
  • School of Computing Science, Simon Fraser University, Canada
Víctor Dalmau
  • Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
Benoît Larose
  • LACIM, Université du Québec a Montréal, Montréal, Canada

Funding

This work was done in part while the first three authors were visiting the Simons Institute for the Theory of Computing at University of California, Berkeley.
  • Briceño, Raimundo: The first author was supported by ERC Starting Grants 678520 and 676970.
  • Bulatov, Andrei A.: This work was supported by an NSERC Discovery grant.
  • Dalmau, Víctor: The third author was supported by MICCIN grant TIN2016-76573-C2-1P and Maria de Maeztu Units of Excellence Programme MDM-2015-0502.
  • Larose, Benoît: The fourth author was supported by an NSERC Discovery grant and FRQNT.

Cite AsGet BibTex

Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose. Dismantlability, Connectedness, and Mixing in Relational Structures. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.29

Abstract

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved - like, for example, dismantlability - and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • relational structure
  • constraint satisfaction problem
  • homomorphism
  • mixing properties
  • Gibbs measure

Metrics

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

Related Versions

References

  1. Dimitris Achlioptas, Paul Beame, and Michael Molloy. Exponential bounds for DPLL below the satisfiability threshold. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 139-140, 2004. Google Scholar
  2. Jung-Chao Ban and Chih-Hung Chang. Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts. Trans. Amer. Math. Soc., 369(12):8389-8407, 2017. Google Scholar
  3. Antar Bandyopadhyay and David Gamarnik. Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures &Algorithms, 33(4):452-479, 2008. Google Scholar
  4. Manuel Bodirsky. The Complexity of Constraint Satisfaction Problems (Invited Talk). In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 2-9, 2015. Google Scholar
  5. Christian Borgs, Jennifer Chayes, László Lovász, Vera T Sós, and Katalin Vesztergombi. Counting graph homomorphisms. In Topics in discrete mathematics, pages 315-371. Springer, 2006. Google Scholar
  6. Mike Boyle, Ronnie Pavlov, and Michael Schraudner. Multidimensional sofic shifts without separation and their factors. Trans. Amer. Math. Soc., 362(9):4617-4653, 2010. Google Scholar
  7. Raimundo Briceño. The topological strong spatial mixing property and new conditions for pressure approximation. Ergodic Theory Dynam. Systems, 38(5):1658-1696, 2018. Google Scholar
  8. Raimundo Briceño and Ronnie Pavlov. Strong spatial mixing in homomorphism spaces. SIAM J. Discrete Math., 31(3):2110-2137, 2017. Google Scholar
  9. Raimundo Briceño, Andrei Bulatov, Víctor Dalmau, and Benoit Larose. Long range actions, connectedness, and dismantlability in relational structures. CoRR, abs/1901.04398, 2019. URL: http://arxiv.org/abs/1901.04398.
  10. Raimundo Briceño, Kevin McGoff, and Ronnie Pavlov. Factoring onto ℤ^d subshifts with the finite extension property. Proc. Amer. Math. Soc., 146(12):5129-5140, 2018. Google Scholar
  11. Graham R. Brightwell and Peter Winkler. Graph homomorphisms and phase transitions. J. Combin. Theory Ser. B, 77(2):221-262, 1999. Google Scholar
  12. Graham R. Brightwell and Peter Winkler. Gibbs measures and dismantlable graphs. J. Combin. Theory Ser. B, 78(1):141-166, 2000. Google Scholar
  13. Graham R. Brightwell and Peter Winkler. Graph homomorphisms and long range action. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 63:29-48, 2004. Google Scholar
  14. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330, 2017. Google Scholar
  15. Andrei A. Bulatov, Víctor Dalmau, Martin Grohe, and Dániel Marx. Enumerating homomorphisms. J. Comput. Syst. Sci., 78(2):638-650, 2012. Google Scholar
  16. Andrei A. Bulatov, Andrei A. Krokhin, and Benoit Larose. Dualities for Constraint Satisfaction Problems. In Complexity of Constraints - An Overview of Current Research Themes [Result of a Dagstuhl Seminar], pages 93-124, 2008. Google Scholar
  17. Tullio Ceccherini-Silberstein and Michel Coornaert. On the density of periodic configurations in strongly irreducible subshifts. Nonlinearity, 25(7):2119, 2012. Google Scholar
  18. Víctor Dalmau, Andrei A. Krokhin, and Benoit Larose. First-Order Definable Retraction Problems for Posets and Reflexive Graph. In 19th IEEE Symposium on Logic in Computer Science (LICS 2004), 14-17 July 2004, Turku, Finland, Proceedings, pages 232-241, 2004. Google Scholar
  19. Rina Dechter. Constraint processing. Morgan Kaufmann, 2003. Google Scholar
  20. Martin Dyer and Catherine Greenhill. The complexity of counting graph homomorphisms. Random Structures &Algorithms, 17(3-4):260-289, 2000. Google Scholar
  21. Martin Dyer, Alistair Sinclair, Eric Vigoda, and Dror Weitz. Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures &Algorithms, 24(4):461-479, 2004. Google Scholar
  22. 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
  23. Jan Foniok, Jaroslav Nesetril, and Claude Tardif. Generalised dualities and maximal finite antichains in the homomorphism order of relational structures. Eur. J. Comb., 29(4):881-899, 2008. Google Scholar
  24. David Gamarnik and Dmitriy Katz. Sequential cavity method for computing free energy and surface pressure. Journal of Statistical Physics, 137(2):205, 2009. Google Scholar
  25. Hans-Otto Georgii. Gibbs Measures and Phase Transitions, volume 9 of De Gruyter Studies in Mathematics. Berlin, 2 edition, 2011. Google Scholar
  26. Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Complexity of Reconfiguration Problems for Constraint Satisfaction. CoRR, abs/1812.10629, 2018. Google Scholar
  27. Pavol Hell and Jaroslav Nešetřil. Graphs and homomorphisms, volume 28 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004. Google Scholar
  28. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011. Google Scholar
  29. David Kelly and Ivan Rival. Crowns, Fences, and Dismantlable Lattices. Canadian Journal of Mathematics, 26(5):1257-1271, 1974. Google Scholar
  30. P. Kolaitis and M. Vardi. Conjunctive-Query Containment and Constraint Satisfaction. Journal of Computer and System Sciences, 61:302-332, 2000. Google Scholar
  31. Marcin Kozik. Weak consistency notions for all the CSPs of bounded width. In 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, pages 633-641, 2016. Google Scholar
  32. Florent Krcommatailzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. PNAS, 104(25):10318-10323, 2007. Google Scholar
  33. Gábor Kun, Ryan O'Donnell, Suguru Tamaki, Yuichi Yoshida, and Yuan Zhou. Linear programming, width-1 CSPs, and robust satisfaction. In Innovations in Theoretical Computer Science, ICT '12, pages 484-495, 2012. Google Scholar
  34. Benoit Larose, Cynthia Loten, and Claude Tardif. A characterisation of first-order constraint satisfaction problems. Log. Methods Comput. Sci., 3(4):4:6, 22, 2007. Google Scholar
  35. Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge university press, 1995. Google Scholar
  36. Douglas Lind and Klaus Schmidt. Symbolic and algebraic dynamical systems. In Handbook of dynamical systems, volume 1, pages 765-812. Elsevier, 2002. Google Scholar
  37. Brian Marcus and Joachim Rosenthal. Codes, systems, and graphical models, volume 123. Springer Science &Business Media, 2012. Google Scholar
  38. Fabio Martinelli, Enzo Olivieri, and Roberto H Schonmann. For 2-D lattice spin systems weak mixing implies strong mixing. Communications in Mathematical Physics, 165(1):33-47, 1994. Google Scholar
  39. Marc Mézard and Andrea Montanari. Information, Physics, and Computation. Oxford University press, 2009. Google Scholar
  40. Naomi Nishimura. Introduction to Reconfiguration. Algorithms, 11(4):52, 2018. Google Scholar
  41. Richard Nowakowski and Peter Winkler. Vertex-to-vertex pursuit in a graph. Discrete Mathematics, 43(2-3):235-239, 1983. Google Scholar
  42. Ronnie Pavlov and Michael Schraudner. Entropies realizable by block gluing ℤ^d shifts of finite type. Journal d'Analyse Mathématique, 126(1):113-174, 2015. Google Scholar
  43. Dror Weitz. Counting Independent Sets Up to the Tree Threshold. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 140-149, New York, NY, USA, 2006. ACM. Google Scholar
  44. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342, 2017. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact