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.

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Abstract-In all-optical networks that use WDM technology it is often the case that several communication requests have to be blocked, due to bandwidth and technology limitations. Minimizing request blocking is therefore an important task calling for algorithmic techniques for efficient routing and wavelength assignment.Here we study the problem for rings under both the undirected and the directed settings, corresponding to symmetric and oneway communication respectively. The problem in graph-theoretic terms can be formulated as the MAXIMUM ROUTING AND PATH COLORING PROBLEM. We present a Chain-and-Matching technique for routing requests and coloring the corresponding paths which gives constant approximations for both the undirected and the directed cases. For the undirected problem we obtain a

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We study deterministic gossiping in ad hoc radio networks with large node labels. The labels (identifiers) of the nodes come from a domain of size N which may be much larger than the size n of the network (the number of nodes). Most of the work on deterministic communication has been done for the model with small labels which assumes N = O(n). A notable exception is Peleg's paper [32], where the problem of deterministic communication in ad hoc radio networks with large labels is raised and a deterministic broadcasting algorithm is proposed, which runs in O(n 2 log n) time for N polynomially large in n. The O(n log 2 n)-time deterministic broadcasting algorithm for networks with small labels given by Chrobak et al.[11] implies deterministic O(n log N log n)-time broadcasting and O(n 2 log 2 N log n)-time gossiping in networks with large labels. We propose two new deterministic gossiping algorithms for ad hoc radio networks with large labels, which are the first such algorithms with subquadratic time for polynomially large N . More specifically, we propose:-a deterministic O(n 3/2 log 2 N log n)-time gossiping algorithm for directed networks; and -a deterministic O(n log 2 N log 2 n)-time gossiping algorithm for undirected networks.