TY - GEN

T1 - From complexity to algebra and back:

T2 - 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015)

AU - Carvalho, Catarina

AU - Madelaine, Florent

AU - Martin, Barnaby

N1 - This is the accepted version of the following article: C. Carvalho, M. Florent, & M. Barnaby, “Fom complexity to algebra and back: digraph classes, collapsibility, and the PGP”, published in Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 6-10 July 2015, IEEE Xplore Digital Library, August 2015.
The final, published version is available online via doi: 10.1109/LICS.2015.50
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PY - 2015/8/3

Y1 - 2015/8/3

N2 - 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.

AB - 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.

KW - algebra

KW - complexity theory

KW - cloning

KW - polynomials

KW - electronic mail

KW - concrete

U2 - 10.1109/LICS.2015.50

DO - 10.1109/LICS.2015.50

M3 - Conference contribution

SN - 1043-6871

SP - 462

EP - 474

BT - Proceedings of the 2015 30th Annual ACM / IEEE Symposium on logic in Computer Science (LICS)

PB - Institute of Electrical and Electronics Engineers (IEEE)

Y2 - 6 July 2015 through 10 July 2015

ER -