Approximation algorithms for path coloring in trees

The study of the path coloring problem is motivated by the allocation of optical bandwidth to communication requests in all-optical networks that utilize Wavelength Division Multiplexing (WDM). WDM technology establishes communication between pairs of network nodes by establishing transmitter- receiver paths and assigning wavelengths to each path so that no two paths going through the same fiber link use the same wavelength. Optical bandwidth is the number of distinct wavelengths. Since state-of-the-art technology allows for a limited number of wavelengths, the engineering problem to be solved is to establish communication minimizing the total number of wavelengths used. This is known as the wavelength routing problem. In the case where the underlying network is a tree, it is equivalent to the path coloring problem. We survey recent advances on the path coloring problem in both undirected and bidirected trees. We present hardness results and lower bounds for the general problem covering also the special case of sets of symmetric paths (corresponding to the important case of symmetric communication). We give an overview of the main ideas of deterministic greedy algorithms and point out their limitations. For bidirected trees, we present recent results about the use of randomization for path coloring and outline approximation algorithms that find path colorings by exploiting fractional path colorings. Also, we discuss upper and lower bounds on the performance of on-line algorithms.