From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP

Carvalho, Catarina; Madelaine, Florent R.; Martin, Barnaby

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.

Citation

Carvalho, C., Madelaine, F. R., & Martin, B. (2015, July). From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP. Presented at 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, Kyoto, Japan

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015
• Start Date: Jul 6, 2015
• End Date: Jul 10, 2015
• Acceptance Date: Mar 30, 2015
• Online Publication Date: Aug 3, 2015
• Publication Date: Aug 3, 2015
• Deposit Date: Oct 14, 2016
• Publicly Available Date: Oct 17, 2016
• Pages: 462-474
• Series ISSN: 1043-6871
• Book Title: Proceedings, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015) : 6-10 July 2015, Kyoto, Japan.
• ISBN: 9781479988471
• DOI: https://doi.org/10.1109/lics.2015.50
• Public URL: https://durham-repository.worktribe.com/output/1149604