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Algebraic Approach to Promise Constraint Satisfaction

Authors:

Andrei Krokhin,

Jakub OpršalAuthors Info & Claims

Journal of the ACM (JACM), Volume 68, Issue 4

Article No.: 28, Pages 1 - 66

https://doi.org/10.1145/3457606

Published: 14 July 2021 Publication History

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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.

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Index Terms

Algebraic Approach to Promise Constraint Satisfaction
1. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

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cover image Journal of the ACM

Journal of the ACM Volume 68, Issue 4

August 2021

297 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3468065

Editor:
Éva Tardos
Cornell University, United States

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Published: 14 July 2021

Accepted: 01 March 2021

Revised: 01 November 2020

Received: 01 July 2019

Published in JACM Volume 68, Issue 4

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