Michael Pinsker

Michael Pinsker, Jakub Rydval, Moritz Schöbi et al.

The Feder-Vardi dichotomy conjecture for Constraint Satisfaction Problems (CSPs) with finite templates, confirmed independently by Bulatov and Zhuk, has an extension to certain well-behaved infinite templates due to Bodirsky and Pinsker which remains wide open. We formulate three fundamental questions on the scope of the Bodirsky-Pinsker conjecture and provide positive answers to them. Our first two main results provide two simplifications of this scope, one of structural, and the other one of algebraic nature. The former simplification implies that the conjecture is equivalent to its restriction to templates without algebraicity, a crucial assumption in the most powerful classification methods. The latter yields that the higher-arity invariants of any template within its scope can be assumed to be essentially injective, and any algebraic condition characterizing any complexity class within the conjecture closed under Datalog reductions must be satisfiable by injections, thus lifting the mystery of the better applicability of certain algebraic conditions over others. Our third main result uses the first one to show that any non-trivially tractable template within the scope serves, up to a Datalog-computable modification of it, as the witness of the tractability of a non-finitely tractable finite-domain Promise Constraint Satisfaction Problem (PCSP) by the so-called sandwich method. This provides a particularly strong connection between the Bodirsky-Pinsker conjecture and finite-domain PCSPs. In the light of the third main result, we initiate a new case study-of phylogeny CSPs-which we investigate from the perspective of descriptive complexity. Within this study, we show that there exists a tractable phylogeny CSP that pp-constructs a finite-domain PCSP inexpressible in fixed-point logic with counting but does not pp-construct any finite-domain CSP with this property.

Johanna Brunar, Marcin Kozik, Tomáš Nagy et al.

Two major milestones on the road to the full complexity dichotomy for finite-domain constraint satisfaction problems were Bulatov's proof of the dichotomy for conservative templates, and the structural dichotomy for smooth digraphs of algebraic length 1 due to Barto, Kozik, and Niven. We lift the combined scenario to the infinite, and prove that any smooth digraph of algebraic length 1 pp-constructs, together with pairs of orbits of an oligomorphic subgroup of its automorphism group, every finite structure -- and hence its conservative graph-colouring problem is NP-hard -- unless the digraph has a pseudo-loop, i.e. an edge within an orbit. We thereby overcome, for the first time, previous obstacles to lifting structural results for digraphs in this context from finite to $ω$-categorical structures; the strongest lifting results hitherto not going beyond a generalisation of the Hell-Nešetřil theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary $ω$-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

Johanna Brunar, Michael Pinsker, Moritz Schöbi

The Bodirsky-Kára classification of temporal constraint languages stands as one of the earliest and most seminal complexity classifications within infinite-domain Constraint Satisfaction Problems (CSPs), yet it remains one of the most mysterious in terms of algorithms and algebraic invariants for the tractable cases. We show that those temporal languages which do not pp-construct EVERYTHING (and thus by the classification are solvable in polynomial time) have, in fact, very limited expressive power as measured by the graphs and hypergraphs they can pp-interpret. This limitation yields many previously unknown algebraic consequences, while also providing new, uniform proofs for known invariance properties. In particular, we show that such temporal constraint languages admit $4$-ary pseudo-Siggers polymorphisms -- a result that sustains the possibility that the existence of such polymorphisms extends to the much broader context of the Bodirsky-Pinsker conjecture.