General Lower Bounds and Improved Algorithms for Infinite–Domain CSPs

Abstract

We study the fine-grained complexity of NP-complete, infinite-domain constraint satisfaction problems (CSPs) parameterised by a set of first-order definable relations (with equality). Such CSPs are of central importance since they form a subclass of any infinite-domain CSP parameterised by a set of first-order definable relations over a relational structure (possibly containing more than just equality). We prove that under the randomised exponential-time hypothesis it is not possible to find $$c > 1$$ c > 1 such that a CSP over an arbitrary finite equality language is solvable in $$O(c^n)$$ O ( c n ) time (n is the number of variables). Stronger lower bounds are possible for infinite equality languages where we rule out the existence of $$2^{o(n \log n)}$$ 2 o ( n log n ) time algorithms; a lower bound which also extends to satisfiability modulo theories solving for an arbitrary background theory. Despite these lower bounds we prove that for each $$c > 1$$ c > 1 there exists an NP-hard equality CSP solvable in $$O(c^n)$$ O ( c n ) time. Lower bounds like these immediately ask for closely matching upper bounds, and we prove that a CSP over a finite equality language is always solvable in $$O(c^n)$$ O ( c n ) time for a fixed c, and manage to extend this algorithm to the much broader class of CSPs where constraints are formed by first-order formulas over a unary structure.