ON THE COMPLEXITY OF COMPUTING OPTIMAL SOLUTIONS

ZHI-ZHONG CHEN, SEINOSUKE TODA
1991 International Journal of Foundations of Computer Science  
We study the computational complexity of computing optimal solutions but not just giving optimal costs for NP optimization problems where the costs of feasible solutions are bounded above by a polynomial in,the length of their instances (we call such an NP optimization problem an NP combinatomal optimization problem, or simply, an NPCOP). It is of particular interest to find a computational structure (or equivalently, a complexity class) which captures that complexity, if we consider the
more » ... s of computing optimal $solutions\cdot for$ NPCOP's as a class of functions giving those optimal solutions. In this paper, we will observe that $PF_{\iota\iota}^{NP}$ , 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 NPCOP $\Pi$ , there exists a polynomial-time bounded randomized algorithm which, given an instance of $\Pi$ , 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 NPCOP's, any function giving those optimal solutions is at least as computationally hard as all functions in $P\Gamma_{\iota\iota}^{\tau^{NP}}$ . To show the hardness results, we introduce a property of NPCOP's, called linear paddability, and we show a general result that if $\Pi$ is a linearly paddable NPCOP and its associated decision problem is NP-hard, then all functions in $P\Gamma_{\iota\iota}^{NP}$ { are computable in polynomial-time with one free evaluation of an arbitrary function giving optimal solutions for instances of $\Pi$ . The hardness results are applications of this general result. Among the NPCOP's, we include MAXIMUM CLIQUE, MINIMUM COLORING, LONGEST PATH, 0-1 TRAVELING SALESPERSON, and 0-1 INTEGER PROGRAMMING. to be NP-complete. Another approach to NPOP's has been developed by Krentel [6] . In [6] , he considered the complexity of NPOP's by investigating the computational complexity of computing optimal costs for NPOP's. He defined two classes of functions to capture that complexity. One is $PF^{NP}$ , the class of functions computable in polynomial-time with polynomial number of queries to NP oracle sets. The other is $PF^{NP[log1}$ , the class of functions computable in polynomial-time with logarithmic number of queries to NP oracle sets. Krentel showed that computing optimal costs for several well-known
doi:10.1142/s0129054191000133 fatcat:exbfirow7rfvvktkecrpd4hs4a