From complexity to algebra and back: digraph classes, collapsibility and the PGP

Catarina Carvalho, Florent Madelaine, Barnaby Martin

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

6 Citations (Scopus)
85 Downloads (Pure)

Abstract

Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idem potent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semi complete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its Pi2-restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete.
Original languageEnglish
Title of host publicationProceedings of the 2015 30th Annual ACM / IEEE Symposium on logic in Computer Science (LICS)
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
Pages462-474
Number of pages13
ISBN (Print)1043-6871
DOIs
Publication statusPublished - 3 Aug 2015
Event30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015) - Kyoto, Japan
Duration: 6 Jul 201510 Jul 2015

Conference

Conference30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015)
Country/TerritoryJapan
CityKyoto
Period6/07/1510/07/15

Keywords

  • algebra
  • complexity theory
  • cloning
  • polynomials
  • electronic mail
  • concrete

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