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Algebraic Approach to Promise Constraint Satisfaction

Barto, Libor; Bulín, Jakub; Krokhin, Andrei; Opršal, Jakub

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Libor Barto

Jakub Bulín

Jakub Opršal


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.


Barto, L., Bulín, J., Krokhin, A., & Opršal, J. (2021). Algebraic Approach to Promise Constraint Satisfaction. Journal of the ACM, 68(4), 1-66.

Journal Article Type Article
Acceptance Date Mar 16, 2021
Online Publication Date Jul 14, 2021
Publication Date 2021-08
Deposit Date Jul 16, 2021
Publicly Available Date Oct 25, 2021
Journal Journal of the ACM
Print ISSN 0004-5411
Electronic ISSN 1557-735X
Publisher Association for Computing Machinery (ACM)
Peer Reviewed Peer Reviewed
Volume 68
Issue 4
Article Number 28
Pages 1-66


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