Robust Approximation of Temporal CSP

A temporal constraint language Γ is a set of relations with first-order definitions in (Q;<). Let CSP(Γ) denote the set of constraint satisfaction problem instances with relations from Γ. CSP(Γ) admits robust approximation if, for any ε ≥ 0, given a (1 − ε)-satisfiable instance of CSP(Γ), we can compute an assignment that satisfies at least a (1−f(ε))-fraction of constraints in polynomial time. Here, f(ε) is some function satisfying f(0) = 0 and lim ε→0 f(ε) = 0. Firstly, we give a qualitative characterization of robust approximability: Assuming the Unique Games Conjecture, we give a necessary and sufficient condition on Γ under which CSP(Γ) admits robust approximation. Secondly, we give a quantitative characterization of robust approximability: Assuming the Unique Games Conjecture, we precisely characterize how f(ε) depends on ε for each Γ. We show that our robust approximation algorithms can be run in almost linear time.