Algorithms with large domination ratio

Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A, n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial-time algorithm with domination ratio 1 - o(1) for the partition problem, and a deterministic, polynomial-time algorithm with domination ratio Ω(1) for the MaxCut problem and for some far-reaching extensions of it, including Max-r-Sat, for each fixed r. The techniques combine combinatorial and probabilistic methods with tools from harmonic analysis.