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Algebraic approach to promise constraint satisfaction.

Bulin, J. and Krokhin, A. and Oprsal, J. (2019) 'Algebraic approach to promise constraint satisfaction.', in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019). New York: ACM, pp. 602-613.


The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 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 paper 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 improving the state-of-the-art in approximate graph colouring: we show that, for any k ≥ 3, it is NP-hard to find a (2k − 1)-colouring of a given k-colourable graph.

Item Type:Book chapter
Full text:(AM) Accepted Manuscript
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Publisher statement:© ACM 2019. This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing,
Date accepted:23 April 2019
Date deposited:26 April 2019
Date of first online publication:23 June 2019
Date first made open access:23 June 2019

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