We study the computational complexity of computing optimal solutions (the solutions themselves, not just their cost) for NP optimization problems where the costs of feasible solutions are bounded above by a polynomial in the length of their instances (we simply denote by NPOP such an NP optimization problem). It is of particular interest to find a computational structure (or equivalently, a complexity class) which. captures that complexity, if we consider the problems of computing optimal solutions for NPOP’s as a class of functions giving those optimal solutions. In this paper, we will observe that [Formula: see text] the class of functions computable in polynomial-time with one free evaluation of unbounded parallel queries to NP oracle sets, captures that complexity. We first show that for any NPOP Π, there exists a polynomial-time bounded randomized algorithm which, given an instance of Π, uses one free evaluation of parallel queries to an NP oracle set and outputs some optimal solution of the instance with very high probability. We then show that for several natural NPOP’s, any function giving those optimal solutions is at least as computationally hard as all functions in [Formula: see text]. To show the hardness results, we introduce a property of NPOP’s, called paddability, and we show a general result that if Π is a paddable NPOP and its associated decision problem is NP-hard, then all functions in [Formula: see text] are computable in polynomial-time with one free evaluation of an arbitrary function giving optimal solutions for instances of Π. The hardness results are applications of this general result. Among the NPOP’s, we include MAXIMUM CLIQUE, MINIMUM COLORING, LONGEST PATH, LONGEST CYCLE, 0–1 TRAVELING SALESPERSON, and 0–1 INTEGER PROGRAMMING.
Structural properties of bounded relations with an application to NP optimization problems