Generalized Majority-Minority Operations are Tractable

Generalized majority-minority (GMM) operations are introduced as a common generalization of near unanimity operations and Mal'tsev operations on finite sets. We show that every instance of the constraint satisfaction problem (CSP), where all constraint relations are invariant under a (fixed) GMM operation, is solvable in polynomial time. This constitutes one of the largest tractable cases of the CSP

Beyond Hypertree Width: Decomposition Methods Without Decompositions

The general intractability of the constraint satisfaction problem has motivated the study of restrictions on this problem that permit polynomial-time solvability. One major line of work has focused on structural restrictions, which arise from restricting the interaction among constraint scopes. In this paper, we engage in a mathematical investigation of generalized hypertree width, a structural measure that has up to recently eluded study. We obtain a number of computational results, including a simple proof of the tractability of CSP instances having bounded generalized hypertree width

Linear Datalog and Bounded Path Duality of Relational Structures

In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem \csp(\best) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL

The Complexity of the Distributed Constraint Satisfaction Problem

We study the complexity of the Distributed Constraint Satisfaction Problem (DCSP) on a synchronous, anonymous network from a theoretical standpoint. In this setting, variables and constraints are controlled by agents which communicate with each other by sending messages through fixed communication channels. Our results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template's invariance under certain operations. Specifically, we show that DCSP($\Gamma$) is polynomial-time tractable if and only if $\Gamma$ is invariant under symmetric polymorphisms of all arities. Otherwise, there are no algorithms that solve DCSP($\Gamma$) in finite time. We also show that the same condition holds for the search variant of DCSP. Collaterally, our results unveil a feature of the processes' neighbourhood in a distributed network, its iterated degree, which plays a major role in the analysis. We explore this notion establishing a tight connection with the basic linear programming relaxation of a CSP.Comment: Full version of a STACS'21 pape

Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used. We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming

Fractional homomorphism, Weisfeiler-Leman invariance, and the Sherali-Adams hierarchy for the Constraint Satisfaction Problem

Given a pair of graphs $\textbf{A}$ and $\textbf{B}$, the problems of deciding whether there exists either a homomorphism or an isomorphism from $\textbf{A}$ to $\textbf{B}$ have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where $\textbf{A}$ and $\textbf{B}$ are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.Comment: Full version of a MFCS'21 pape

The Product Homomorphism Problem and Applications

The product homomorphism problem (PHP) takes as input a finite collection of structures A_1, ..., A_n and a structure B, and asks if there is a homomorphism from the direct product between A_1, A_2, ..., and A_n, to B. We pinpoint the computational complexity of this problem. Our motivation stems from the fact that PHP naturally arises in different areas of database theory. In particular, it is equivalent to the problem of determining whether a relation is definable by a conjunctive query, and the existence of a schema mapping that fits a given collection of positive and negative data examples. We apply our results to obtain complexity bounds for these problems

Conjunctive Queries: Unique Characterizations and Exact Learnability

We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts

Dismantlability, connectedness, and mixing in relational structures

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Brice\~no. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.Comment: 27 pages, full version of the paper accepted to ICALP 201