Complexity Classification Transfer for CSPs via Algebraic Products
Manuel Bodirsky, Peter Jönsson, Barnaby D. Martin, Antoine Mottet, Žaneta Semanišinová · SIAM Journal on Computing · 2024
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.