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Topology and adjunction in promise constraint satisfaction (2023)
Journal Article
Krokhin, A., Opršal, J., Wrochna, M., & Živný, S. (2023). Topology and adjunction in promise constraint satisfaction. SIAM Journal on Computing, 52(1), 38-79. https://doi.org/10.1137/20m1378223

The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a c-colouring of a graph that is promised to be k-colourable, where c≥k. This problem naturally generalises to promise graph homomorphism problems... Read More about Topology and adjunction in promise constraint satisfaction.

Algebraic Approach to Promise Constraint Satisfaction (2021)
Journal Article
Barto, L., Bulín, J., Krokhin, A., & Opršal, J. (2021). Algebraic Approach to Promise Constraint Satisfaction. Journal of the ACM, 68(4), 1-66. https://doi.org/10.1145/3457606

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 appr... Read More about Algebraic Approach to Promise Constraint Satisfaction.

Robust algorithms with polynomial loss for near-unanimity CSPs (2019)
Journal Article
Dalmau, V., Kozik, M., Krokhin, A., Makarychev, K., Makarychev, Y., & Oprsal, J. (2019). Robust algorithms with polynomial loss for near-unanimity CSPs. SIAM Journal on Computing, 48(6), 1763-1795. https://doi.org/10.1137/18m1163932

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a xed domain to the variables so that all constraints are satised. In the optimizatio... Read More about Robust algorithms with polynomial loss for near-unanimity CSPs.

Towards a characterization of constant-factor approximable Finite-Valued CSPs (2018)
Journal Article
Dalmau, V., Krokhin, A., & Manokaran, R. (2018). Towards a characterization of constant-factor approximable Finite-Valued CSPs. Journal of Computer and System Sciences, 97, 14-27. https://doi.org/10.1016/j.jcss.2018.03.003

We study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. Ene et al. have shown that, under a mild technical condition... Read More about Towards a characterization of constant-factor approximable Finite-Valued CSPs.

Binarisation for Valued Constraint Satisfaction Problems (2017)
Journal Article
Cohen, D., Cooper, M., Jeavons, P., Krokhin, A., Powell, R., & Zivny, S. (2017). Binarisation for Valued Constraint Satisfaction Problems. SIAM Journal on Discrete Mathematics, 31(4), 2279-2300. https://doi.org/10.1137/16m1088107

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. S... Read More about Binarisation for Valued Constraint Satisfaction Problems.

The complexity of general-valued CSPs (2017)
Journal Article
Kolmogorov, V., Krokhin, A., & Rolínek, M. (2017). The complexity of general-valued CSPs. SIAM Journal on Computing, 46(3), 1087-1110. https://doi.org/10.1137/16m1091836

An instance of the valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values spec... Read More about The complexity of general-valued CSPs.

Polymorphisms, and how to use them (2017)
Book Chapter
Barto, L., Krokhin, A., & Willard, R. (2017). Polymorphisms, and how to use them. In A. Krokhin, & S. Živný (Eds.), The constraint satisfaction problem : complexity and approximability (1-44). Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/dfu.vol7.15301.1

This article describes the algebraic approach to Constraint Satisfaction Problem that led to many developments in both CSP and universal algebra. No prior knowledge of universal algebra is assumed.

The complexity of Valued CSPs (2017)
Book Chapter
Krokhin, A., & Živný, S. (2017). The complexity of Valued CSPs. In A. Krokhin, & S. Živný (Eds.), The constraint satisfaction problem : complexity and approximability (233-266). Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/dfu.vol7.15301.9

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify gen... Read More about The complexity of Valued CSPs.

On algebras with many symmetric operations (2016)
Journal Article
Carvalho, C., & Krokhin, A. (2016). On algebras with many symmetric operations. International Journal of Algebra and Computation, 26(05), 1019-1032. https://doi.org/10.1142/s0218196716500429

An n-ary operation f is called symmetric if, for all permutations π of {1, . . . , n}, it satisfies the identity f(x1, x2, . . . , xn) = f(xπ(1), xπ(2), . . . , xπ(n) ). We show that, for each finite algebra A, either it has symmetric term operations... Read More about On algebras with many symmetric operations.

Characterizations of several Maltsev conditions (2015)
Journal Article
Kozik, M., Krokhin, A., Valeriote, M., & Willard, R. (2015). Characterizations of several Maltsev conditions. Algebra Universalis, 73(3), 205-224. https://doi.org/10.1007/s00012-015-0327-2

ame congruence theory identifies six Maltsev conditions associated with locally finite varieties omitting certain types of local behaviour. Extending a result of Siggers, we show that of these six Maltsev conditions only two of them are equivalent to... Read More about Characterizations of several Maltsev conditions.

Towards a Characterization of Constant-Factor Approximable Min CSPs (2014)
Book Chapter
Dalmau, V., Krokhin, A., & Manokaran, R. (2015). Towards a Characterization of Constant-Factor Approximable Min CSPs. In P. Indyk (Ed.), Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms : San Diego, California, USA, January 4-6, 2015 (847-857). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973730.58

We study the approximability of Minimum Constraint Satisfaction Problems (Min CSPs) with a fixed finite constraint language Γ on an arbitrary finite domain. The goal in such a problem is to minimize the number of unsatisfied constraints in a given in... Read More about Towards a Characterization of Constant-Factor Approximable Min CSPs.

Oracle Tractability of Skew Bisubmodular Functions (2014)
Journal Article
Huber, A., & Krokhin, A. (2014). Oracle Tractability of Skew Bisubmodular Functions. SIAM Journal on Discrete Mathematics, 28(4), 1828-1837. https://doi.org/10.1137/130936038

In this paper we consider skew bisubmodular functions as recently introduced by the authors and Powell. We construct a convex extension of a skew bisubmodular function which we call Lovász extension in correspondence to the submodular case. We use th... Read More about Oracle Tractability of Skew Bisubmodular Functions.

The Complexity of Valued Constraint Satisfaction (2014)
Journal Article
Jeavons, P., Krokhin, A., & Živný, S. (2014). The Complexity of Valued Constraint Satisfaction. Bulletin of the European Association for Theoretical Computer Science, 113, 21-55

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general fea... Read More about The Complexity of Valued Constraint Satisfaction.

Skew Bisubmodularity and Valued CSPs (2014)
Journal Article
Huber, A., Krokhin, A., & Powell, R. (2014). Skew Bisubmodularity and Valued CSPs. SIAM Journal on Computing, 43(3), 1064-1084. https://doi.org/10.1137/120893549

An instance of the (finite-)valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of values, and a sum of (rational-valued) functions, with each function depending on a subset of the variables. The goal... Read More about Skew Bisubmodularity and Valued CSPs.

Robust Satisfiability for CSPs: Hardness and Algorithmic Results (2013)
Journal Article
Dalmau, V., & Krokhin, A. (2013). Robust Satisfiability for CSPs: Hardness and Algorithmic Results. ACM Transactions on Computation Theory, 5(4), Article 15. https://doi.org/10.1145/2540090

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least a (1 − f(ε))-fraction of constraints for each (1 − ε)-satisfiable instance (i.e., such that at most a ε-fraction of constraints needs... Read More about Robust Satisfiability for CSPs: Hardness and Algorithmic Results.