On the Complexity of the Regenerator Cost Problem in General Networks with Traffic Grooming

We consider the problem of minimizing the number of regenerators in optical networks with traffic grooming. In this problem we are given a network with an underlying topology of a graph G, a set of requests that correspond to paths in G and two positive integers g and d. There is a need to put a regenerator every d edges of every path, because of a degradation in the quality of the signal. Each regenerator can be shared by at most g paths, g being the grooming factor. On the one hand, we show that even in the case of d = 1 the problem is APX - hard, i.e. a polynomial time approximation scheme for it does not exist (unless P = NP). On the other hand, we solve such a problem for general G and any d and g, by providing an 0(log g)-approximation algorithm and thus extending previous results holding only for specific topologies and specific values of d or g.