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

TL;DR

This paper explores the algebraic properties of certain digraph classes, establishing dichotomy theorems related to polymorphisms and their implications for the complexity of the QCSP, including a new trichotomy result.

Contribution

It proves algebraic dichotomies for digraph classes, studies collapsibility and PGP, and introduces a new QCSP complexity trichotomy with constants.

Findings

Partially reflexive paths have polymorphisms with PGP or EGP.

Semicomplete digraphs exhibit similar polymorphism properties.

With constants, QCSPs on paths have complexities in NL, NP-complete, or Pspace-complete.

Abstract

Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idempotent 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 idempotent 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 semicomplete 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 $\Pi_2$-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.