Algebraic approach to promise constraint satisfaction

Jakub Bulín, Andrei Krokhin, Jakub Opršal

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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.
Original languageEnglish
Title of host publicationSTOC 2019
Subtitle of host publicationProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
EditorsMoses Charikar, Edith Cohen
PublisherAssociation for Computing Machinery (ACM)
Pages602–613
Number of pages12
ISBN (Print)9781450367059
DOIs
Publication statusPublished - 23 Jun 2019
EventSTOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing - Phoenix, United States
Duration: 23 Jun 201926 Jun 2019

Conference

ConferenceSTOC '19
Abbreviated titleSTOC 2019
Country/TerritoryUnited States
CityPhoenix
Period23/06/1926/06/19

Keywords

  • constraint satisfaction
  • promise problem
  • approximation
  • graph colouring
  • polymorphism

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