On the power of counting the total number of computation paths of NPTMs

Complexity classes defined by modifying the acceptance condition of NP computations have been extensively studied. For example, the class UP, which contains decision problems solvable by non-deterministic polynomial-time Turing machines (NPTMs) with at most one accepting path -- equivalently NP problems with at most one solution -- has played a significant role in cryptography, since P=/=UP is equivalent to the existence of one-way functions. In this paper, we define and examine variants of several such classes where the acceptance condition concerns the total number of computation paths of an NPTM, instead of the number of accepting ones. This direction reflects the relationship between the counting classes #P and TotP, which are the classes of functions that count the number of accepting paths and the total number of paths of NPTMs, respectively. The former is the well-studied class of counting versions of NP problems, introduced by Valiant (1979). The latter contains all self-reducible counting problems in #P whose decision version is in P, among them prominent #P-complete problems such as Non-negative Permanent, #PerfMatch, and #Dnf-Sat, thus playing a significant role in the study of approximable counting problems. We show that almost all classes introduced in this work coincide with their '# accepting paths'-definable counterparts. As a result, we present a novel family of complete problems for the classes parity-P, Modkp, SPP, WPP, C=P, and PP that are defined via TotP-complete problems under parsimonious reductions.Comment: 19 pages, 1 figur

Routing and Wavelength Assignment in Multifiber WDM Networks with Non-Uniform Fiber Cost

Motivated by the increasing importance of multifiber WDM networks we study a routing and wavelength assignment problem in such networks. In this problem the number of wavelengths per fiber is given and the goal is to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is non-uniform, that is the cost of hiring a fiber may di#er from link to link

Self-Reducibility of Hard Counting Problems withDecisionVersioninP ⋆

Abstract. Many NP-complete problems have counting versions which are #P-complete. On the other hand, #Perfect Matchings is also Cook-complete for #P, which is surprising as Perfect Matching is actually in P (which implies that #Perfect Matchings cannot be Karp-complete for #P). Here, we study the complexity class #PE (functions of #P with easy decision version). The inclusion #PE ⊆ #P is proper unless P = NP. Several natural #PE problems (e.g., #Perfect Matchings, #DNF-Sat, #NonCliques) are shown to possess a specific self-reducibility property. This implies membership in class TotP [KPSZ98,PZ05]. We conjecture that all non-trivial problems of #PE share this self-reducibility property.