Random sampling and approximation of MAX-CSP problems

We present a new efficient sampling method for approximating r-dimensional Maximum Constraint Satisfaction Problems, MAX-rCSP, on n variables up to an additive error εn^r. We prove a new general paradigm in that it suffices, for a given set of constraints, to pick a small uniformly random subset of its variables, and the optimum value of the subsystem induced on these variables gives (after a direct normalization and with high probability) an approximation to the optimum of the whole system up to an additive error of εn^r. Our method gives for the first time a polynomial in ε^-1 bound on the sample size necessary to carry out the above approximation. Moreover, this bound is independent in the exponent on the dimension r. The above method gives a completely uniform sampling technique for all the MAX-rCSP problems, and improves the best known sample bounds for the low dimensional problems, like MAX-CUT. The method of solution depends on a new result on the cut norm of random subarrays, and a new sampling technique for high dimensional linear programs. This method could be also of independent interest.