# Lower Bounds and Faster Algorithms for Equality Constraints

# Lower Bounds and Faster Algorithms for Equality Constraints

## Peter Jonsson, Victor Lagerkvist

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence

Main track. Pages 1784-1790.
https://doi.org/10.24963/ijcai.2020/247

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. We prove that under the randomised exponential-time hypothesis it is not possible to find c > 1 such that a CSP over an arbitrary finite equality language is solvable in 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) 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 there exists an NP-hard equality CSP solvable in 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) time for a fixed c.

Keywords:

Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Constraints and SAT: Constraint Satisfaction

Constraints and SAT: Satisfiability Modulo Theories