Larrauri and Živný [ICALP’24/ACM ToCL’24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid \({N}\) assuming that a solution exists over a monoid \({M}\) , where both monoids are finite and \({M}\) admits a homomorphism to \({N}\) . Using the algebraic approach to promise constraint satisfaction problems, we extend their complexity classification in two directions: we obtain a complexity dichotomy in the case where arbitrary relations are added to the monoids, and we moreover allow the monoid \({M}\) to be finitely generated.

We establish a framework that allows us to transfer results between some constraint satisfaction problems with infinite templates and promise constraint satisfaction problems. On the one hand, we obtain new algebraic results for infinite-domain CSPs giving new criteria for NP-hardness. On the other hand, we show the existence of promise CSPs with finite templates that reduce naturally to tractable infinite-domain CSPs in the scope of the Bodirsky-Pinsker conjecture, but that are not finitely tractable, thereby showing a non-trivial connection between those two fields of research. In an important part of our proof, we also obtain uniform polynomial-time algorithms solving temporal constraint satisfaction problems.

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongrácz (ICALP'16) on graph satisfiability.

We introduce the novel machinery of smooth approximations to provide a systematic algebraic approach to the complexity of CSPs over finitely bounded homogeneous structures. We apply smooth approximations to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament and to give new short proofs of the conjecture for various homogeneous graphs including the random graph (STOC’11, ICALP’16, JACM 2015, SICOMP 2019), and for expansions of the order of the rationals (STOC’08, JACM 2009). Apart from obtaining these dichotomy results, we show how our new proof technique allows one to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize for the first time those CSPs that are solvable by local consistency methods, again using the same machinery.

Abstract. We study the complexity of infinite-domain constraint satisfaction problems (CSPs): our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure [Formula: see text] can be transferred to a classification of the CSPs of first-order expansions of another structure [Formula: see text]. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the [Formula: see text]-fold algebraic power of [Formula: see text]. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of [Formula: see text] and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen’s Interval Algebra, the [Formula: see text]-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyze with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the artificial intelligence (AI) literature.

Abstract. We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities [Formula: see text]) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain constraint satisfaction problems (CSPs) studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of Monotone Monadic SNP (MMSNP) sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al. In particular, the bounded width hierarchy collapses in those cases as well. Our results extend the scope of theorems of Barto and Kozik characterizing bounded width for finite structures and show the applicability of infinite-domain CSPs to other fields.

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.

We introduce the novel machinery of smooth approximations, and apply it to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament, and to give new short proofs of the conjecture for various homogeneous graphs including the random graph (STOC’11, ICALP’16), and for expansions of the order of the rationals (STOC’08). Apart from obtaining these dichotomy results, we show how our new proof technique allows to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize for the first time those CSPs which are solvable by local consistency methods, again using the same machinery.

We produce a class of $\omega$-categorical structures with finite signature by applying a model-theoretic construction---a refinement of the Hrushovski-encoding---to $\omega$-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate $\omega$-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity and $\omega$-categorical templates that show that membership in any given complexity class containing AC$^0$ cannot be expressed by a set of identities on the polymorphisms. It moreover enables us to prove that recent results about the relevance of topology on polymorphism clones of $\omega$-categorical structures also apply for CSP templates, i.e., structures in a finite language. Finally, we obtain a concrete algebraic criterion which could constitute a description of the delineation between tractability and NP-hardness in the dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures.

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well.

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a natural algebraic question: which finite algebras admit only polynomially many homomorphisms into them?We give some sufficient and some necessary conditions for a finite algebra to have this property. In particular, we show that every finite equationally nontrivial algebra has this property which gives us, as a simple consequence, a complete complexity classification of CSPs over two-element structures, thus extending the classification for two-element relational structures by Schaefer (STOC'78).We also present examples of two-element structures that have bounded width but do not have relational width (2,3), thus demonstrating that, from a descriptive complexity perspective, allowing operations leads to a richer theory.

Abstract We prove that if an $\omega $ -categorical structure has an $\omega $ -categorical homogeneous Ramsey expansion, then so does its model-complete core.

The logic MMSNP is a restricted fragment of existential second-order logic which can express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi, who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs that does not rely on the results of Kun. The new universal-algebraic proof allows us to obtain a stronger statement and to verify the more general Bodirsky--Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite $\omega$-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property.

We produce a class of ω-categorical structures with finite signature by applying a model-theoretic construction - a refinement of an encoding due to Hrushosvki - to ω-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate ω-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity, and ω-categorical templates that show that membership in any given complexity class cannot be expressed by a set of identities on the polymorphisms. It moreover enables us to prove that recent results about the relevance of topology on polymorphism clones of ω-categorical structures also apply for CSP templates, i.e., structures in a finite language. Finally, we obtain a concrete algebraic criterion which could constitute a description of the delineation between tractability and NP-hardness in the dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures.

Abstract The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit0was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite setGof ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of nontrivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of nontrivial height 1 identities differ for ω -categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

We investigate the complexity of the containment problem "Does L(A)subseteq L(B) hold?", where B is an unambiguous register automaton and A is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when B is restricted to have a fixed number of registers.

We study formalisms for temporal and spatial reasoning in the modern context of Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pp-homotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen's Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.

The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs that does not rely on the results of Kun. This new proof allows us to obtain a stronger statement and to verify the Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite ω-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP.

We study the Constraint Satisfaction Problem CSP( A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) over the set of integers whose constraint language consists of relations that are first-order definable over the order of the integers. We prove that every discrete temporal CSP is in P or NP-complete, unless it can be formulated as a finite domain CSP, in which case the computational complexity is not known in general.

Many natural decision problems can be formulated as constraint satisfaction problems for reducts $\mathbb{A}$ of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of the topological polymorphism clone of $\mathbb{A}$. Moreover, we study the subclass $\mathcal{C}$ of CSPs for structures $\mathbb{A}$ that are reducts of a structure with a unary language. Also this class $\mathcal{C}$ properly extends the class of all finite-domain CSPs. We apply our new tractability conditions to prove the general tractability conjecture of Bodirsky and Pinsker for reducts of finitely bounded homogeneous structures for the class $\mathcal{C}$.

Many natural decision problems can be formulated as constraint satisfaction problems for reducts of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of topological polymorphism clones. Moreover, we study the subclass C of CSPs for structures that are first-order definable over equality with parameters. Also this class C properly extends the class of all finite-domain CSPs. We show that the tractability conjecture for reducts of finitely bounded homogeneous structures is for C equivalent to the finite-domain tractability conjecture.