Wavelength conversion in optical networks

In many models of optical routing, we are given a set of communication paths in a network, and we must assign a wavelength to each path so that paths sharing an edge receive different wavelengths. The goal is to assign as few wavelengths as possible, in order to make as efficient use as possible of the optical bandwidth. Much work in the area of optical networks has considered the use of wavelength converters: if a node of a network contains a wavelength converter, any path that passes through this node may change its wavelength. Having converters at some of the nodes can reduce the number of wavelengths required for routing, down to the following natural congestion bound: even with converters, we will always need at least as many wavelengths as the maximum number of paths sharing a single edge. Thus Wilfong and Winkler defined a set S of nodes in a network to be sufficient if, placing converters at the nodes in S, every set of paths can be routed with a number of wavelengths equal to its congestion bound. They showed that finding a sufficient set of minimum size is NP-complete, even in planar graphs. In this paper, we provide a polynomial-time algorithm to find a sufficient set for an arbitrary directed network whose size is within a factor of 2 of minimum. We also observe that improving on the factor of 2 would lead to a corresponding improvement for the vertex cover problem. For the case of planar graphs, we provide a polynomial-time approximation scheme. The algorithms are based on connections between the minimum sufficient set problem and the undirected feedback vertex set problem. In particular, as a component of the algorithm on planar graphs, we develop the first polynomial-time approximation scheme for the undirected feedback vertex set problem in planar graphs, a result that we feel to be of interest in its own right.