Open Access The purpose of reoptimization using approximation methods—application of knowledge about the solution of the initial instance I, provided to achieve a better quality of approximation (approximation ratio) of an algorithm for determining optimal or close to it solutions of some “minor” changes of instance I. To solve the problem Ins-Max-EkCSP-P (reoptimization of Max-EkCSP-P with the addition of one constraint) with approximation resistant predicate P exists a polynomial threshold (optimal)
-approximation algorithm,
where 
the threshold “random” approximation ratio of P). When the unique games conjecture (UGC) is hold there exists a polynomial threshold (optimal) 
-approximation algorithm (where 
and 
the integrality gap of semidefinite relaxation of Max-EkCSP-P problem Z) to solve the problem Ins-Max-EkCSP-P.