The Time Complexity of Constraint Satisfaction

We study the time complexity of (d, k)-CSP, the problem of deciding satisfiability of a constraint system C with n variables, domain size d, and at most k variables per constraint. We are interested in the question how the domain size d influences the complexity of deciding satisfiability. We show, assuming the Exponential Time Hypothesis, that two special cases, namely (d, 2)-CSP with bounded variable frequency and d-UNIQUE-CSP, already require exponential time Ω(dc·n) for some c > 0 independent of d. UNIQUE-CSP is the special case for which it is guaranteed that every input constraint system has at most 1 satisfying assignment.