Libor Barto
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Abstract
The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the past 20 years. A new version of the CSP, the promise CSP (PCSP), has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms—high-dimensional symmetries of solution spaces—to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases.
The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this article, we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a “measure of symmetry” that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by giving both general and specific hardness/tractability results. Among other things, we improve the state-of-the-art in approximate graph colouring by showing that, for any k≥ 3, it is NP-hard to find a (2k-1)-colouring of a given k-colourable graph.
- Erhard Aichinger and Peter Mayr. 2016. Finitely generated equational classes. J. Pure Appl. Algeb. 220, 8 (2016), 2816–2827. DOI:https://doi.org/10.1016/j.jpaa.2016.01.001.Google Scholar
Cross Ref
- Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk. 1997. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54, 2 (Apr. 1997), 317–331. DOI:https://doi.org/10.1006/jcss.1997.1472. Google ScholarDigital Library
- Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. 1998. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (1998), 501–555. Google ScholarDigital Library
- Sanjeev Arora and Shmuel Safra. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1 (Jan. 1998), 70–122. DOI:https://doi.org/10.1145/273865.273901. Google ScholarDigital Library
- Per Austrin, Amey Bhangale, and Aditya Potukuchi. 2019. Simplified inapproximability of hypergraph coloring via -agreeing families. (Apr. 2019). arXiv:1904.01163.Google Scholar
- Per Austrin, Amey Bhangale, and Aditya Potukuchi. 2020. Improved inapproximability of rainbow coloring. In Proceedings of the 31st ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 1479–1495. DOI:https://doi.org/10.1137/1.9781611975994.90. Google ScholarDigital Library
- Per Austrin, Venkatesan Guruswami, and Johan Håstad. 2017. -sat is NP-hard. SIAM J. Comput. 46, 5 (2017), 1554–1573. DOI:https://doi.org/10.1137/15M1006507.Google ScholarCross Ref
- Per Austrin and Johan Håstad. 2013. On the usefulness of predicates. ACM Trans. Comput. Theor. 5, 1 (2013), 1–24. Google ScholarDigital Library
- Libor Barto. 2013. Finitely related algebras in congruence distributive varieties have near unanimity terms. Canad. J. Math. 65, 1 (2013), 3–21.Google ScholarCross Ref
- Libor Barto. 2019. Promises make finite (constraint satisfaction) problems infinitary. In Proceedings of the 34th ACM/IEEE Symposium on Logic in Computer Science (LICS’19). IEEE, 1–8. Google ScholarDigital Library
- Libor Barto, Michael Kompatscher, Miroslav Olšák, Trung Van Pham, and Michael Pinsker. 2017. The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems. In Proceedings of the 32nd ACM/IEEE Symposium on Logic in Computer Science (LICS’17). 1–12. DOI:https://doi.org/10.1109/LICS.2017.8005128. Google ScholarDigital Library
- Libor Barto and Marcin Kozik. 2012. Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Meth. Comput. Sci. 8, 1:07 (2012), 1–26. DOI:https://doi.org/10.2168/LMCS-8(1:7)2012.Google ScholarCross Ref
- Libor Barto and Marcin Kozik. 2014. Constraint satisfaction problems solvable by local consistency methods. J. ACM 61, 1 (Jan. 2014), 3:1–3:19. DOI:https://doi.org/10.1145/2556646. Google ScholarDigital Library
- Libor Barto and Marcin Kozik. 2016. Robustly solvable constraint satisfaction problems. SIAM J. Comput. 45, 4 (2016), 1646–1669. DOI:https://doi.org/10.1137/130915479.Google ScholarDigital Library
- Libor Barto, Andrei Krokhin, and Ross Willard. 2017. Polymorphisms, and how to use them. In The Constraint Satisfaction Problem: Complexity and Approximability, Andrei Krokhin and Stanislav Živný (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 1–44. DOI:https://doi.org/10.4230/DFU.Vol7.15301.1.Google Scholar
- Libor Barto, Jakub Opršal, and Michael Pinsker. 2018. The wonderland of reflections. Israel J. Math. 223, 1 (Feb. 2018), 363–398. DOI:https://doi.org/10.1007/s11856-017-1621-9.Google ScholarCross Ref
- Libor Barto and Michael Pinsker. 2020. Topology is irrelevant (in a dichotomy conjecture for infinite domain constraint satisfaction problems). SIAM J. Comput. 49, 2 (2020), 365–393. DOI:https://doi.org/10.1137/18M1216213.Google ScholarDigital Library
- Mihir Bellare, Oded Goldreich, and Madhu Sudan. 1998. Free bits, PCPs, and nonapproximability—Towards tight results. SIAM J. Comput. 27, 3 (1998), 804–915. DOI:https://doi.org/10.1137/S0097539796302531. Google ScholarDigital Library
- Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. 1988. Multi-prover interactive proofs: How to remove intractability assumptions. In Proceedings of the 20th ACM Symposium on Theory of Computing (STOC’88). ACM, New York, NY, 113–131. DOI:https://doi.org/10.1145/62212.62223. Google ScholarDigital Library
- Manuel Bodirsky. 2008. Constraint satisfaction problems with infinite templates. In Complexity of Constraints (2009-05-05) (Lecture Notes in Computer Science), Nadia Creignou, Phokion G. Kolaitis, and Heribert Vollmer (Eds.), Vol. 5250. Springer, 196–228. DOI:https://doi.org/10.1007/978-3-540-92800-3_8.Google Scholar
- Manuel Bodirsky and Martin Grohe. 2008. Non-dichotomies in constraint satisfaction complexity. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP’08). 184–196. Google ScholarDigital Library
- Manuel Bodirsky and Marcello Mamino. 2017. Constraint satisfaction problems over numeric domains. In The Constraint Satisfaction Problem: Complexity and Approximability, Andrei Krokhin and Stanislav Živný (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 79–111. DOI:https://doi.org/10.4230/DFU.Vol7.15301.79.Google Scholar
- Manuel Bodirsky, Antoine Mottet, Miroslav Olšák, Jakub Opršal, Michael Pinsker, and Ross Willard. 2019. Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems). In Proceedings of the 34th ACM/IEEE Symposium on Logic in Computer Science (LICS’19). IEEE, 1–12. Google ScholarDigital Library
- V. G. Bodnarchuk, L. A. Kaluzhnin, Viktor N. Kotov, and Boris A. Romov. 1969. Galois theory for Post algebras. I. Cybernetics 5, 3 (1969), 243–252.Google ScholarCross Ref
- V. G. Bodnarchuk, L. A. Kaluzhnin, Viktor N. Kotov, and Boris A. Romov. 1969. Galois theory for Post algebras. II. Cybernetics 5, 5 (1969), 531–539.Google ScholarCross Ref
- Joshua Brakensiek and Venkatesan Guruswami. 2016. New hardness results for graph and hypergraph colorings. In Proceedings of the 31st Conference on Computational Complexity (CCC’16) (Leibniz International Proceedings in Informatics (LIPIcs)), Ran Raz (Ed.), Vol. 50. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 14:1–14:27. DOI:https://doi.org/10.4230/LIPIcs.CCC.2016.14. Google ScholarDigital Library
- Joshua Brakensiek and Venkatesan Guruswami. 2016. Promise constraint satisfaction: Algebraic structure and a symmetric Boolean dichotomy. ECCC Report No. 183 (2016). Retrieved from https://eccc.weizmann.ac.il/report/2016/183/.Google Scholar
- Joshua Brakensiek and Venkatesan Guruswami. 2017. The quest for strong inapproximability results with perfect completeness. In Proceedings of the Approximation, Randomization, and Combinatorial Optimization Conference: Algorithms and Techniques (APPROX/RANDOM’17). 4:1–4:20. DOI:https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.4.Google Scholar
- Joshua Brakensiek and Venkatesan Guruswami. 2018. Promise constraint satisfaction: Structure theory and a symmetric Boolean dichotomy. In Proceedings of the 29th ACM-SIAM Symposium on Discrete Algorithms (SODA’18). Society for Industrial and Applied Mathematics, Philadelphia, PA, 1782–1801. DOI:https://doi.org/10.1137/1.9781611975031.117. Google ScholarDigital Library
- Joshua Brakensiek and Venkatesan Guruswami. 2019. An algorithmic blend of LPs and ring equations for promise CSPs. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA’19). 436–455. DOI:https://doi.org/10.1137/1.9781611975482.28. Google ScholarDigital Library
- Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, and Stanislav Živný. 2020. The power of the combined basic LP and affine relaxation for promise CSPs. SIAM J. Comput. 49, 6 (2020), 1232–1248. DOI:https://doi.org/10.1137/20M1312745 arXiv:1907.04383v3.Google ScholarDigital Library
- Jonah Brown-Cohen and Prasad Raghavendra. 2015. Combinatorial Optimization Algorithms via Polymorphisms. (Jan. 2015). arXiv:1501.01598.Google Scholar
- Andrei Bulatov and Peter Jeavons. 2001. Algebraic Structures in Combinatorial Problems. Technical Report. Technische Universität Dresden.Google Scholar
- Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. 2005. Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 3 (Mar. 2005), 720–742. DOI:https://doi.org/10.1137/S0097539700376676. Google ScholarDigital Library
- Andrei A. Bulatov. 2013. The complexity of the counting constraint satisfaction problem. J. ACM 60, 5 (Oct. 2013). Google ScholarDigital Library
- Andrei A. Bulatov. 2017. A dichotomy theorem for nonuniform CSPs. In Proceedings of the IEEE 58th Symposium on Foundations of Computer Science (FOCS’17). 319–330. DOI:https://doi.org/10.1109/FOCS.2017.37.Google ScholarCross Ref
- Andrei A. Bulatov, Andrei Krokhin, and Benoit Larose. 2008. Dualities for constraint satisfaction problems. In Complexity of Constraints: An Overview of Current Research Themes. Springer Berlin, 93–124. DOI:https://doi.org/10.1007/978-3-540-92800-3_5. Google ScholarDigital Library
- Jakub Bulín, Andrei Krokhin, and Jakub Opršal. 2019. Algebraic approach to promise constraint satisfaction. In Proceedings of the 51st ACM SIGACT Symposium on the Theory of Computing (STOC’19). ACM, New York, NY, 602–613. DOI:https://doi.org/10.1145/3313276.3316300. Google ScholarDigital Library
- Hubie Chen and Benoit Larose. 2017. Asking the metaquestions in constraint tractability. ACM Trans. Comput. Theor. 9, 3 (Oct. 2017), 11:1–11:27. DOI:https://doi.org/10.1145/3134757. Google ScholarDigital Library
- Hubie Chen, Matthew Valeriote, and Yuichi Yoshida. 2016. Testing assignments to constraint satisfaction problems. In Proceedings of the IEEE 57th Symposium on Foundations of Computer Science (FOCS’16). 525–534. DOI:https://doi.org/10.1109/FOCS.2016.63.Google ScholarCross Ref
- Víctor Dalmau, Marcin Kozik, Andrei A. Krokhin, Konstantin Makarychev, Yury Makarychev, and Jakub Opršal. 2017. Robust algorithms with polynomial loss for near-unanimity CSPs. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA’17). 340–357. DOI:https://doi.org/10.1137/1.9781611974782.22. Google ScholarDigital Library
- Victor Dalmau, Andrei Krokhin, and Rajsekar Manokaran. 2018. Towards a characterization of constant-factor approximable finite-valued CSPs. J. Comput. System Sci. 97 (2018), 14–27.Google ScholarCross Ref
- Víctor Dalmau and Justin Pearson. 1999. Closure functions and width 1 problems. In Principles and Practice of Constraint Programming – CP’99, Joxan Jaffar (Ed.). Springer Berlin, 159–173. Google ScholarDigital Library
- Irit Dinur. 2007. The PCP theorem by gap amplification. J. ACM 54, 3 (June 2007). DOI:https://doi.org/10.1145/1236457.1236459. Google ScholarDigital Library
- Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. 2005. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34, 5 (May 2005), 1129–1146. DOI:https://doi.org/10.1137/S0097539704443057. Google ScholarDigital Library
- Irit Dinur, Elchanan Mossel, and Oded Regev. 2009. Conditional hardness for approximate coloring. SIAM J. Comput. 39, 3 (2009), 843–873. DOI:https://doi.org/10.1137/07068062X. Google ScholarDigital Library
- Irit Dinur, Oded Regev, and Clifford Smyth. 2005. The hardness of 3-uniform hypergraph coloring. Combinatorica 25, 5 (Sept. 2005), 519–535. DOI:https://doi.org/10.1007/s00493-005-0032-4. Google ScholarDigital Library
- Tomás Feder and Moshe Y. Vardi. 1998. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28, 1 (Feb. 1998), 57–104. DOI:https://doi.org/10.1137/S0097539794266766. Google ScholarDigital Library
- Miron Ficak, Marcin Kozik, Miroslav Olšák, and Szymon Stankiewicz. 2019. Dichotomy for symmetric Boolean PCSPs. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP’19) (Leibniz International Proceedings in Informatics (LIPIcs)), Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi (Eds.), Vol. 132. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 57:1–57:12. DOI:https://doi.org/10.4230/LIPIcs.ICALP.2019.57 arXiv:1904.12424.Google Scholar
- M. R. Garey and David S. Johnson. 1976. The complexity of near-optimal graph coloring. J. ACM 23, 1 (1976), 43–49. DOI:https://doi.org/10.1145/321921.321926. Google ScholarDigital Library
- David Geiger. 1968. Closed systems of functions and predicates. Pacific J. Math. 27 (1968), 95–100. Google ScholarCross Ref
- Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, and Girish Varma. 2017. Super-polylogarithmic hypergraph coloring hardness via low-degree long codes. SIAM J. Comput. 46, 1 (2017), 132–159. DOI:https://doi.org/10.1137/140995520.Google ScholarCross Ref
- Venkatesan Guruswami and Sanjeev Khanna. 2004. On the hardness of 4-coloring a 3-colorable graph. SIAM J. Disc. Math. 18, 1 (2004), 30–40. DOI:https://doi.org/10.1137/S0895480100376794. Google ScholarDigital Library
- Venkatesan Guruswami and Euiwoong Lee. 2017. Strong inapproximability results on balanced rainbow-colorable hypergraphs. Combinatorica (14 Dec. 2017). DOI:https://doi.org/10.1007/s00493-016-3383-0. Google ScholarDigital Library
- Venkatesan Guruswami and Sai Sandeep. 2020. -to-1 hardness of coloring 3-colorable graphs with colors. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP’20) (Leibniz International Proceedings in Informatics (LIPIcs)), Artur Czumaj, Anuj Dawar, and Emanuela Merelli (Eds.), Vol. 168. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 62:1–62:12. DOI:https://doi.org/10.4230/LIPIcs.ICALP.2020.62.Google Scholar
- Johan Håstad. 2001. Some optimal inapproximability results. J. ACM 48, 4 (July 2001), 798–859. DOI:https://doi.org/10.1145/502090.502098. Google ScholarDigital Library
- Pavol Hell and Jaroslav Nešetřil. 1990. On the complexity of -coloring. J. Combin. Theor. Ser. B 48, 1 (1990), 92–110. DOI:10.1016/0095-8956(90)90132-J
Google ScholarDigital Library
- Pavol Hell and Jaroslav Nešetřil. 2004. Graphs and Homomorphisms. Oxford University Press.Google Scholar
- Sangxia Huang. 2013. Improved hardness of approximating chromatic number. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, and José D. P. Rolim (Eds.). Springer, Berlin, 233–243. DOI:https://doi.org/10.1007/978-3-642-40328-_17.Google ScholarCross Ref
- Paweł M. Idziak, Petar Marković, Ralph McKenzie, Matthew Valeriote, and Ross Willard. 2010. Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 7 (2010), 3023–3037. DOI:https://doi.org/10.1137/090775646. Google ScholarDigital Library
- Peter Jeavons. 1998. On the algebraic structure of combinatorial problems. Theor. Comput. Sci. 200, 1–2 (1998), 185–204. Google ScholarDigital Library
- Peter Jeavons, David Cohen, and Marc Gyssens. 1997. Closure properties of constraints. J. ACM 44, 4 (July 1997), 527–548. DOI:https://doi.org/10.1145/263867.263489. Google ScholarDigital Library
- Ken-ichi Kawarabayashi and Mikkel Thorup. 2017. Coloring 3-colorable graphs with less than colors. J. ACM 64, 1 (2017), 4:1–4:23. DOI:https://doi.org/10.1145/3001582.Google Scholar
- Sanjeev Khanna, Nathan Linial, and Shmuel Safra. 2000. On the hardness of approximating the chromatic number. Combinatorica 20, 3 (01 Mar. 2000), 393–415. DOI:https://doi.org/10.1007/s004930070013.Google Scholar
- Subhash Khot. 2001. Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science. IEEE, 600–609. Google ScholarDigital Library
- Phokion G. Kolaitis and Moshe Y. Vardi. 2008. A logical approach to constraint satisfaction. In Complexity of Constraints: An Overview of Current Research Themes, Nadia Creignou, Phokion G. Kolaitis, and Heribert Vollmer (Eds.). LNCS, Vol. 5250. Springer-Verlag, 125–155. Google ScholarDigital Library
- Vladimir Kolmogorov, Andrei Krokhin, and Michal Rolínek. 2017. The complexity of general-valued CSPs. SIAM J. Comput. 46, 3 (2017), 1087–1110.Google ScholarDigital Library
- Andrei Krokhin and Jakub Opršal. 2019. The complexity of 3-colouring -colourable graphs. In Proceedings of the IEEE 60th Symposium on Foundations of Computer Science (FOCS’19). 1227–1239. DOI:https://doi.org/10.1109/FOCS.2019.00076 arxiv:1904.03214.Google ScholarCross Ref
- Andrei Krokhin, Jakub Opršal, Marcin Wrochna, and Stanislav Živný. 2020. Topology and adjunction in promise constraint satisfaction. (Mar. 2020). arXiv:2003.11351.Google Scholar
- Andrei Krokhin and Stanislav Živný (Eds.). 2017. The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.Google Scholar
- Gábor Kun, Ryan O’Donnell, Suguru Tamaki, Yuichi Yoshida, and Yuan Zhou. 2012. Linear programming, width-1 CSPs, and robust satisfaction. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference (ITCS’12). ACM, New York, NY, 484–495. DOI:https://doi.org/10.1145/2090236.2090274. Google ScholarDigital Library
- Gábor Kun and Mario Szegedy. 2016. A new line of attack on the dichotomy conjecture. Eur. J. Comb. 52 (2016), 338–367. Google ScholarDigital Library
- Benoit Larose. 2017. Algebra and the complexity of digraph CSPs: A survey. In The Constraint Satisfaction Problem: Complexity and Approximability, Andrei Krokhin and Stanislav Živný (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 267–285. DOI:https://doi.org/10.4230/DFU.Vol7.15301.267.Google Scholar
- Benoit Larose and László Zádori. 2007. Bounded width problems and algebras. Algeb. Univers. 56, 3 (01 June 2007), 439–466. DOI:https://doi.org/10.1007/s00012-007-2012-6.Google Scholar
- Lászlo Lovász. 1978. Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theor. Ser. A 25, 3 (1978), 319–324. DOI:https://doi.org/10.1016/0097-3165(78)90022-5.Google ScholarCross Ref
- Colin McDiarmid. 1993. A random recolouring method for graphs and hypergrams. Combinat. Proba. Comput. 2 (1993), 363–365. DOI:https://doi.org/10.1017/S0963548300000730.Google ScholarCross Ref
- Walter D. Neumann. 1974. On Malcev conditions. J. Austral. Math. Soc. 17, 3 (5 1974), 376–384. DOI:https://doi.org/10.1017/S1446788700017122.Google ScholarCross Ref
- Miroslav Olšák. 2017. The weakest nontrivial idempotent equations. Bull. London Math. Soc. 49, 6 (2017), 1028–1047. DOI:https://doi.org/10.1112/blms.12097.Google ScholarCross Ref
- Miroslav Olšák. 2018. Personal communication.Google Scholar
- Miroslav Olšák. 2019. Loop conditions. Algeb. Univers. 81, 1 (Nov. 2019), 2:1–2:11. DOI:https://doi.org/10.1007/s00012-019-0631-3 arXiv:1701.00260.Google Scholar
- Jakub Opršal. 2017. Taylor’s modularity conjecture and related problems for idempotent varieties. Order (Nov. 2017). DOI:https://doi.org/10.1007/s11083-017-9441-4.Google Scholar
- Michael Pinsker. 2015. Algebraic and model theoretic methods in constraint satisfaction. (2015). arXiv:1507.00931.Google Scholar
- Nicholas Pippenger. 2002. Galois theory for minors of finite functions. Disc. Math. 254, 1 (2002), 405–419. DOI:https://doi.org/10.1016/S0012-365X(01)00297-7.Google ScholarCross Ref
- Ran Raz. 1998. A parallel repetition theorem. SIAM J. Comput. 27, 3 (1998), 763–803. DOI:https://doi.org/10.1137/S0097539795280895. Google ScholarDigital Library
- Omer Reingold. 2008. Undirected connectivity in log-space. J. ACM 55, 4 (Sept. 2008), 17:1–17:24. DOI:https://doi.org/10.1145/1391289.1391291. Google ScholarDigital Library
- Thomas J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th ACM Symposium on Theory of Computing (STOC’78). ACM, New York, NY, 216–226. DOI:https://doi.org/10.1145/800133.804350. Google ScholarDigital Library
- Alexander Schrijver. 1978. Vertex-critical subgraphs of Kneser graphs. Nieuw Arch. Wisk. (3) 26, 3 (1978), 454–461. Google Scholar
- Mark H. Siggers. 2010. A strong Mal’cev condition for locally finite varieties omitting the unary type. Algeb. Univers. 64, 1 (Oct. 2010), 15–20. DOI:https://doi.org/10.1007/s00012-010-0082-3.Google Scholar
- Walter Taylor. 1973. Characterizing Mal’cev conditions. Algeb. Univers. 3 (1973), 351–397. Google ScholarCross Ref
- Johan Thapper and Stanislav Živný. 2016. The complexity of finite-valued CSPs. J. ACM 63, 4 (2016), 37:1–37:33. DOI:https://doi.org/10.1145/2974019. Google ScholarDigital Library
- Marcin Wrochna and Stanislav Živný. 2020. Improved hardness for -colourings of -colourable graphs. In Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 1426–1435. DOI:https://doi.org/10.1137/1.9781611975994.86 arXiv:1907.00872. Google ScholarDigital Library
- Dmitriy Zhuk. 2017. A proof of CSP dichotomy conjecture. In Proceedings of the IEEE 58th Symposium on Foundations of Computer Science (FOCS’17). 331–342. DOI:https://doi.org/10.1109/FOCS.2017.38.Google ScholarCross Ref
- Dmitriy Zhuk. 2020. A proof of the CSP dichotomy conjecture. J. ACM 67, 5 (Aug. 2020), 30:1–30:78. DOI:https://doi.org/10.1145/3402029.Google ScholarDigital Library
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