The complexity of counting functions with easy decision version

Abstract

We investigate the complexity of counting problems that belong to the complexity class #P and have an easy decision version. These problems constitute the class #PE which has some well-known representatives such as #PERFECT MATCHINGS, #DNF-SAT, and NONNEGATIVE PERMANENT. An important property of these problems is that they are all #P-complete, in the Cook sense, while they cannot be #P-complete in the Karp sense unless P = NP. We study these problems in respect to the complexity class TotP, which contains functions that count the number of all paths of a PNTM. We first compare TotP to #P and #PE and show that FP ⊆ TotP ⊆ #PE ⊆ #P and that the inclusions are proper unless P = NP. We then show that several natural #PE problems -including the ones mentioned above -belong to TotP. Moreover, we prove that TotP is exactly the Karp closure of self-reducible functions of #PE. Therefore, all these problems share a remarkable structural property: for each of them there exists a polynomial-time nondeterministic Turing machine which has as many computation paths as the output value. © Springer-Verlag Berlin Heidelberg 2006.