The Complexity of Quantified Constraints: Collapsibility, Switchability, and the Algebraic Formulation

Abstract

Let \(\mathbb {A} \) be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if \(\mathbb {A} \) satisfies the polynomially generated powers property (PGP) and \(\mathcal {B} \) is a constraint language invariant under \(\mathbb {A} \) (that is, in \(\mathrm{Inv}(\mathbb {A}) \) ), then QCSP \((\mathcal {B}) \) is in NP. In doing this we study the special forms of PGP, switchability and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if \(\mathrm{Inv}(\mathbb {A}) \) satisfies the exponentially generated powers property (EGP), then QCSP \((\mathrm{Inv}(\mathbb {A})) \) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen in [1] as a generalisation of the already-known collapsibility [5]. There, an algebra \(\mathbb {A}:=(\lbrace 0,1,2\rbrace ;r) \) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv \((\mathbb {A}) \) , Pol( Δ ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras

“…Application of these techniques became known as the algebraic approach to the CSP, although one may argue that the name misses the point a little -the success of the approach lies mostly in combining and moving back and forth between the relational and algebraic side, and this is the case for this paper as well. The general theory of the CSP was further refined in subsequent papers [31,10] and turned out to be an efficient tool in other types of constraint problems including the Quantified CSP [17,34,62], the Counting CSP [30,29], some optimization problems, e.g. the Valued CSP [51] and robust approximability [7], infinite-domain CSPs [15,14], related promise problems such as "approximate coloring" and the Promise CSP [20,2], and many others.…”