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

Carvalho, Catarina and Madelaine, Florent R. and Martin, Barnaby (2015) 'From complexity to algebra and back : digraph classes, collapsibility, and the PGP.', in Proceedings, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015) : 6-10 July 2015, Kyoto, Japan. Piscataway, NJ: IEEE, pp. 462-474.


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:Book chapter
Full text:(AM) Accepted Manuscript
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Publisher statement:© 2015 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.
Date accepted:30 March 2015
Date deposited:17 October 2016
Date of first online publication:03 August 2015
Date first made open access:No date available

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