Fine-Grained Time Complexity of Constraint Satisfaction Problems

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation SD such that the time complexity of the NP-complete problem CSP({SD}) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({SD}) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.