Polynomial-time optimization, parallel approximation, and fixpoint logic

A study of polynomial-time optimization from the perspective of descriptive complexity theory is initiated. It is established that the class of polynomial-time and polynomially bounded optimization problems with ordered finite structures as instances can be characterized in terms of the stage functions of positive first-order formulas, i.e., the functions that compute the number of distinct stages in the bottom-up evaluation of the least fixpoints of such formulas. After this, the stage functions of several first-order formulas whose least fixpoints form natural p-complete problems are studied, and it is shown that they are not NC-approximate within any factor of the optimum, unless P=NC. Finally, it is proved that certain polynomial-time optimization problems are complete with respect to a new kind of restricted reductions that preserve parallel approximability and are definable using quantifier-free formulae