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

Carvalho, Catarina, Madelaine, Florent R. and Martin, Barnaby (2015) From complexity to algebra and back: digraph classes, collapsibility and the PGP. 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 6-10 Jul 2015, Kyoto, Japan. . ISSN 1043-6871 [Conference or Workshop Item]


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.

Item Type: Conference or Workshop Item (Paper)
Research Areas: A. > School of Science and Technology > Computer Science > Foundations of Computing group
Item ID: 16769
Notes on copyright: © 2016 IEEE.Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
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Depositing User: Barnaby Martin
Date Deposited: 03 Jun 2015 11:32
Last Modified: 24 Apr 2018 11:31

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