Path Multicoloring with Fewer Colors in Spiders and Caterpillars Aris Pagourtzis Katerina Potika Stathis Zachos Department of Computer Science School of Electrical and Computer Engineering National Technical University of Athens email:{pagour,epotik,zachos}@cs.ntua.gr Abstract We study a recently introduced path coloring problem with appli- cations to wavelength assignment in all-optical networks with multiple fibers. In contrast to classical path coloring, it is, in this setting, pos- sible to assign a color more than once to paths that pass through the same edge; the number of allowed repetitions per edge is given and the goal is to minimize the number of colors used. We present algorithms and hardness results for tree topologies of special interest. Our algorithms achieve approximation ratio of 2 in spiders and 3 in caterpillars, whereas the best algorithm for trees so far, achieves an approximation ratio of 4. We also study the directed version of the problem and show that it admits a 3-approximation algorithm in caterpillars, while it can be solved exactly in spiders. AMS Subject Classifications: 68U05,68R10, 05C05, 05C15, 90B18, 05C85, 90C59, 94C15. keywords: path coloring, multifiber all-optical networks, approxima- tion algorithms, spiders, caterpillars. 1 Introduction Optical fibers play an important role in modern communication systems, such as the Internet and other data transferring networks, since they offer a large bandwidth, small delays and low bit-error rates. A network is called all-optical, if signals remain in optical form (no conversion to electrical form) from one end to the other end. The technology that enables transmitting 1