Abstract
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 been shown 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 Briceño. 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.
Highlights
The Constraint Satisfaction Problem (CSP) provides a powerful framework in a wide range of areas of mathematics, computer science, statistical physics, and elsewhere
The CSP appears in different forms: as the standard one outlined above in AI and computer science [19], as the homomorphism problem in graph and model theory [22, 27], as conjunctive query evaluation in logic and database theory [30], as computing the partition function of a spin system in statistical physics [39] and related areas, like symbolic dynamics and coding [35, 37]
The CSP allows for many approaches of diverse nature, and every application field exploits some of its many facets: structural properties of constraints for complexity and algorithms, probabilistic properties and the topology of the solution space in Random CSP and random structures, mixing properties in statistical physics and dynamical systems, decay of correlations and the uniqueness of probabilistic measures in approximate counting, and homomorphic duality and logical characterizations in model theory
Summary
The Constraint Satisfaction Problem (CSP) provides a powerful framework in a wide range of areas of mathematics, computer science, statistical physics, and elsewhere. The particular case when this threshold is 1 has been intensively studied, motivated initially by the fact that the connectedness of the solution space for SAT problems over random instances is linked to the performance of standard satisfiability algorithms, such as WalkSAT or DPLL [1, 32] This has given rise to a general framework called reconfiguration [28] (see [40] for a recent survey) that goes way beyond homomorphisms. Mixing properties have been intensively studied in statistical physics and related areas (see [2, 6, 8, 10, 17, 42]), and are usually applied when the set of particles in G is very large or infinite In this case, it can be very useful to be able to “glue” together partial homomorphisms, provided their domains are far from each other. They can be found at the full version [9]
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