Logical Definability of NP Optimization Problems

Logical Definability of NP Optimization Problems

We investigate here NP optimization problems from a logical deenability standpoint. We show that the class of optimization problems whose optimum is deenable using rst-order formulae coincides with the class of polynomially bounded NP optimization problems on nite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this fram...

Logical Definability of NP-Optimization Problems with Monadic Auxiliary Predicates

Given a rst{order formula ' with predicate symbols e 1 such a solution is deened to be jS 0 j. 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 s 1 ; : : : ; s r are restricted to be monadic. The result is proved by an Ehrenfeucht{Fra ss e game and remains true in several more...

Weighted NP Optimization Problems: Logical Definability and Approximation Properties

Extending a well-known property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characterization. It is shown that any NP optimization problem can be stated as a problem in which the cons...

Weighted NP Optimization Problems: Logical De nability and Approximation Properties

Logical Definability of Counting Functions

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is #P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, #L is the class of functions on nite structures counting the tuples (T ; c) satisfying a given formula (T ; c) in ...