Libor Barto


Symmetry in Computational Complexity (CoCoSym)

Project funded by the European Research Council (ERC) as an ERC Consolidator Grant (ERC CoG), No. 771005

Principal investigator: Libor Barto

PhD students:
  • Diego Battistelli, 1 Oct 2018 - 23 Jan 2020
  • Kristina Asimi since 1 Dec 2018

Host institution: Charles University, Prague, Czech Republic.
Department: Department of Algebra, Faculty of Mathematics and Physics
Dates: 1 Feb 2018 - 31 Jan 2023
Openings: contact me


  1. A. Mottet, Smooth Approximations and CSPs over finitely bounded homogeneous structures, submitted [arXiv]
  2. L. Barto, Z. Brady, A. Bulatov, M. Kozik, D. Zhuk, Minimal Taylor Algebras as a Common Framework for the Three Algebraic Approaches to the CSP, submitted [old preprint]
  3. L. Barto and W. DeMeo, A. Mottet, Constraint Satisfaction Problems over Finite Structures, submitted [arXiv]
  4. K. Asimi, L. Barto, Finitely Tractable Promise Constraint Satisfaction Problems, in preparation [arXiv]
  5. A. Moorhead, Higher Kiss terms, submitted [arXiv]
  6. L. Barto, J. Bulin, A. Krokhin, J. Oprsal, Algebraic approach to promise constraint satisfaction, submitted [arXiv]
  7. L. Barto, D. Battistelli, K. M. Berg, Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case, to appear at STACS 2021 [arXiv]

  8. A. Mottet, M. Pinsker, Cores over Ramsey structures, Journal of Symbolic Logic (online) [arXiv] [DOI]
  9. A. Mottet, K. Quaas, The Containment Problem for Unambiguous Register Automata and Unambiguous Timed Automata, Theory of Computing Systems (2020) [DOI]

  10. M. Bodirsky, A. Mottet, M. Olsak, J. Oprsal, M. Pinsker, R. Willard, omega-categorical structures avoiding heigh 1 identities, Transactions of the AMS, 374 (2021), 327-350 [arXiv] [DOI]
  11. D. Zhuk, A Proof of the CSP Dichotomy Conjecture, Journal of the ACM, 67/5 (2020), 30:1-30:78 [arXiv] [DOI] [Presburger Award 2020]
  12. D. Zhuk, B. Martin, QCSP monsters and the demise of the Chen Conjecture, STOC 2020 [arXiv] [DRO] [DOI]
  13. P. Gillibert, J. Jonušas, M. Kompatscher, A. Mottet, M. Pinsker, Hrushovski’s Encoding and ω-Categorical CSP Monsters, ICALP 2020 [arXiv] [DOI]
  14. L. Barto, M. Kozik, J. Tan, M. Valeriote, Sensitive instances of the Constraint Satisfaction Problem, ICALP 2020 [arXiv] [DOI]
  15. L. Barto, M. Pinsker, Topology is irrelevant (in the infinite domain dichotomy conjecture for constraint satisfaction problems), SIAM Journal on Computing, 49/2 (2020), 365-393 [arXiv] [DOI]
  16. L. Barto, Accessible set endofunctors are universal, Commentationes Mathematicae Universitatis Carolinae 60/4 (2019), 497-508 [arXiv]
  17. A. Mottet, K. Quaas, The containment problem for Unambiguous Register Automata, STACS'19 [arXiv] [DOI]
  18. M. Bodirsky, A. Mottet, M. Olšák, J. Opršal, M. Pinsker, R. Willard, Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems), LICS 2019 [arXiv] [DOI]
  19. L. Barto, Promises make finite (constraint satisfaction) problems infinitary, LICS 2019 [arXiv] [DOI]
  20. L. Barto, Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps, FCT'19, 3-17. (invited talk, non-refereed) [arXiv] [DOI]

Conferences, workshops, seminars

  1. Feb 11 2021, L. Barto, talk Finitely tractable PCSPs at Ulam Seminar, CU Boulder, USA. [Slides] [Video]
  2. Feb 5-7 2021, L. Barto, invited talk Algebras with polynomially many homomorphisms at AAA 100, Krakow, Poland (online). [Slides]
  3. Jan 4,5 2021, L. Barto, A. Mottet, Homogeneous structures: model theory meets universal algebra, Oberwolfach, Germany (online).
    • L. Barto: 2 tutorial lectures Universal Algebra: Tutorial [Slides]
    • A. Mottet: Smooth Approximations [Slides] [Video]
  4. Oct 7 2020, L. Barto, talk CSPs and Symmetries, 19th Jarnik lecture, Prague, Czechia. [Slides]
  5. Sep 20-26 2020, K. Asimi, L. Barto, K. Berg, A. Mottet, W. DeMeo CSP World Congress 2020, Voels am Schlern, Italy.
  6. Aug 6 2020, A. Mottet, talk Cores over Ramsey structures at Midsummer Combinatorial Workshop XXIV, Prague, Czechia.
  7. Jul 23 2020, L. Barto, talk Baby PCP Theorem and reductions between Promise CSPs at Online CSP seminar. [Slides] [Video]
  8. Jul 8-11 2020, L. Barto, talk Sensitive instances of the Constraint Satisfaction Problem at ICALP 2020, online. [Slides] [Video]
  9. Feb 21-23 2020, K. Asimi, L. Barto, K. Berg AAA99, Siena, Italy.
    • K. Asimi: Infinity: Relevant or Irrelevant? [Slides]
    • L. Barto: Reconstructing subproducts from projections [Slides]
    • K. Berg: The Complexity of Homomorphism Factorization [Slides]
  10. Jan 27-31 2020, L. Barto, invited talk Promise Constraint Satisfaction at ATCAGC 2020: Algebraic, Topological and Complexity Aspects of Graph Covers , Bedrichov, Czechia. [Slides]
  11. Jan 12-17, A. Mottet, Ackermann Award 2019 invited talk Dichotomies in constraint satisfaction: canonical functions and numeric CSPs at CSL 2020, Barcelona, Spain. [Slides]
  12. Oct 3-5 2019, K. Asimi, talk Promises turn finite problems to infinite ones at Congress of Young Mathematicians, Novi Sad, Serbia. [Slides (Serbian)]
  13. Sep 16-20 2019, L. Barto, invited talk Promise Constraint Satisfaction Problem at TbiLLC 2019, Tsikhisdziri, Georgia. [Slides]
  14. Sep 1-7 2019, K. Asimi, L. Barto, D. Battistelli, A. Mottet 57th Summer School on Algebra and Ordered Sets, Karolinka, Czechia.
    • K. Asimi: Infinite nature of finite PCSPs [Slides]
    • L. Barto: A hardness criterion for Promise CSPs [Slides]
    • D. Battistelli: On the complexity of symmetric Promise CSP [Slides]
    • A. Mottet: Topology is relevant in infinite-domain constraint satisfaction [Slides]
  15. Aug 11-14 2019, L. Barto, invited talk Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps at FCT, Copenhagen, Denmark. [Slides]
  16. Jun 25-28 2019: D. Battistelli Second Algebra Week, Siena, Italy.
  17. Jun 24-27 2019: L. Barto, A. Mottet LICS 2019, Vancouver, Canada.
    • L. Barto: Promises Make Finite (Constraint Satisfaction) Problems Infinitary [Slides]
    • A. Mottet: Topology is relevant (in a dichotomy conjecture for infinite-domain CSPs) [Slides]
  18. Jun 21-23 2019: K. Asimi, D. Battistelli, D. Zhuk AAA98, Dresden, Germany.
    • D. Zhuk, The complexity of the Quantified Constraint Satisfaction Problem on a 3-element set [Slides]
  19. Jun 17-21 2019, L. Barto, tutorial Constraint Satisfaction Problems at Caleidoscope : Complexity as a Kaleidoscope, Paris, France. [Slides, part I] [Slides, part II]
  20. Jun 9-11 2019: K. Asimi, A. Mottet, D. Zhuk Russian Workshop on Complexity and Model Theory, Moscow, Russia.
    • D. Zhuk, The complexity of the Quantified Constraint Satisfaction Problem on a 3-element set [Slides]
  21. May 10 2019: L. Barto, invited talk Algebraic theory of promise constraint satisfaction problems at Prague Gathering of Logicians 2019, Prague, Czechia. [Slides]
  22. Apr 4-7 2019: K. Asimi, D. Battistelli, A. Mottet, Spring School of the Department of Algebra, Sec-Ustupky, Czechia.
    • K. Asimi, Infinity is Relevant
    • D. Battistelli, Ultrafilters and Compactness of Topological Spaces
  23. Mar 13-16 2019: A. Mottet, talk The containment problem for unambiguous register automata at STACS 2019, Berlin, Germany. [Slides]
  24. Mar 1-3 2019: K. Asimi, L. Barto, D. Battistelli, A. Mottet, D. Zhuk AAA97, Vienna, Austria.
    • L. Barto, A loop lemma for nonidempotent cores [Slides]
    • A. Mottet, Topology is relevant [Slides]
    • D. Zhuk, An exponential lower bound on the size of primitive positive definition [Slides]
  25. Oct 17-23 2018: A. Mottet, invitational workshop Unifying themes in Ramsey theory, Banff, Canada
  26. Sep 2-7 2018: L. Barto, member of the organizing quadrumvirate, [56th Summer School on Algebra and Ordered Sets], Spindleruv Mlyn, Czechia.
  27. Jun 4-8 2018: L. Barto, talk at invitational workshop Cyclic operations in promise constraint satisfaction problems at Dagstuhl workshop The Constraint Satisfaction Problem: Complexity and Approximability, Schloss Dagstuhl, Germany. [Slides]
  28. Jun 1-3 2018: L. Barto, invited talk Height one identities at 96. Arbeitstagung Allgemeine Algebra, Darmstadt, Germany. [Slides]


  1. Aug 25-Sep 8 2020: Jakub Opršal (co-funded by GAČR 18-20123S)
  2. Feb 17-20 2020: Marcin Kozik
  3. Dec 9-13 2019: Marcin Kozik
  4. Nov 17-26 2019: Dmitriy Zhuk
  5. Oct 23-29 2019 Ralph McKenzie
  6. Jul 7-12 2019: Andrei Krokhin and Jakub Opršal
  7. Mar 3-7 2019: Jakub Opršal
  8. Mar 3-8 2019: Andrei Krokhin
  9. Jan 28 - Feb 1 2019: Marcin Kozik
  10. Dec 17-22 2018: Marcin Kozik
  11. Oct 15-26 2018: Marcin Kozik
  12. Mar 26- Apr 6 2018: Andrei Krokhin


  1. Dec 12-17 2019: A. Mottet visiting Michael Pinsker at TU Wien (seminar talk "Cores of structures of Ramsey expansion")
  2. Nov 10-17 2019: A. Moorhead visiting Erhard Aichinger and his group at JKU Linz (seminar talks, including "What is the right definition of supernilpotence")
  3. Oct 20-26 2019: A. Mottet visiting J. Thapper, P. Koiran, and others at University Paris-Est, University Paris 7, ENS Lyon (seminar talk "Symmetries in infinite-domain constraint satisfaction." at each of the universities)
  4. Oct 6-7 2019: L. Barto visiting Michal Botur at Palacky University (seminar talk "Mathematics in coloring problems with promises")
  5. Jun 17-21 2019: A. Mottet meeting Jakub Opršal and Barnaby Martin at TU Dresden
  6. Feb 10-16 2019: L. Barto visiting Marcin Kozik and Miroslav Olsak at Jagiellonian University
  7. May 6-11 2018: L. Barto visiting Andrei Krokhin at Durham University
  8. Feb 23 - Mar 5 2018: L. Barto visiting Marcin Kozik at Jagiellonian University


The last 20 years of rapid development in the computational-theoretic aspects of the fixed-language Constraint Satisfaction Problems (CSPs) has been fueled by a connection between the complexity and a certain concept capturing symmetry of computational problems in this class.

My vision is that this connection will eventually evolve into the organizing principle of computational complexity and will lead to solutions of fundamental problems such as the Unique Games Conjecture or even the P-versus-NP problem. In order to break through the current limits of this algebraic approach, I will concentrate on specific goals designed to

  1. discover suitable objects capturing symmetry that reflect the complexity in problem classes, where such an object is not known yet;
  2. make the natural ordering of symmetries coarser so that it reflects the complexity more faithfully;
  3. delineate the borderline between computationally hard and easy problems;
  4. strengthen characterizations of existing borderlines to increase their usefulness as tools for proving hardness and designing efficient algorithm; and
  5. design efficient algorithms based on direct and indirect uses of symmetries.
The specific goals concern the fixed-language CSP over finite relational structures and its generalizations to infinite domains (∞CSP) and weighted relations (vCSP), in which the algebraic theory is highly developed and the limitations are clearly visible.

The approach is based on joining the forces of the universal algebraic methods in finite domains, model-theoretical and topological methods in the ∞CSP, and analytical and probabilistic methods in the vCSP. The starting point is to generalize and improve the Absorption Theory from finite domains.