We study the problem of absolute approximability of MAX-CSP problems with the global constraints. We prove existence of an efficient sampling method for the MAX-CSP class of problems with linear global constraints and bounded feasibility gap. It gives for the first time a polynomial in epsilon^-1 sample complexity bound for that class of problems. The method yields also the best up to date sample complexity bound for the balanced MAX-CSP problems such as the graph and hypergraph BISECTION problems.