Logical definability of NP-optimisation problems with monadic auxiliary predicates

Abstract

Given a first-order formula ϕ with predicate symbols e1…el, so,…,sr, an NP-optimisation problem on -structures can be defined as follows: for every -structure G, a sequence of relations on G is a feasible solution iff satisfies φ, and the value of such a solution is defined to be ׀S0׀. In a strong sense, every polynomially bounded NP-optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1,…,sr are restricted to be monadic. The result is proved by an Ehrenfeucht-Fraïssé game and remains true in several more general situations.