Complexity of counting the optimal solutions
Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #.C for any complexity class C of decision problems. In particular, the classes#. Pi(k)P with k >= 1 corresponding to all levels of the polynomial hierarchy, have thus been studied. However, for a large variety of counting problems arising from optimization problems, a precise complexity classification turns out to be impossible with these classes. In order to remedy this unsatisfactory situation, we introduce a hierarchy of new counting complexity classes # . Opt(k)P and # . Opt(k)P[log n] with k >= 1. We prove several important properties of these new classes, like closure properties and the relationship with the # . Pi(k)P-classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes. (C) 2009 Elsevier B.V. All rights reserved.
counting complexity optimization problems computational complexity