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Robust Approximation of Temporal CSP

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