Approximating the Traffic Grooming Problem with Respect to ADMs and OADMs

Abstract

We consider the problem of switching cost in optical networks, where messages are sent along lightpaths. Given lightpaths, we have to assign them colors, so that at most glightpaths of the same color can share any edge (gis the grooming factor). The switching of the lightpaths is performed by electronic ADMs (Add-Drop-Multiplexers) at their endpoints and optical ADMs (OADMs) at their intermediate nodes. The saving in the switching components becomes possible when lightpaths of the same color can use the same switches. Whereas previous studies concentrated on the number of ADMs, we consider the cost function - incurred also by the number of OADMs - of f(i¾?) = i¾?|OADMs| + (1 i¾? i¾?)|ADMs|, where 0 ≤ i¾?≤ 1. We concentrate on chain networks, but our technique can be directly extended to ring networks. We show that finding a coloring which will minimize this cost function is NP-complete, even when the network is a chain and the grooming factor is g= 2, for any value of i¾?. We then present a general technique that, given an r-approximation algorithm working on particular instances of our problem, i.e. instances in which all requests share a common edge of the chain, builds a new algorithm for general instances having approximation ratio ri¾?logni¾?. This technique is used in order to obtain two polynomial time approximation algorithms for our problem: the first one minimizes the number of OADMs (the case of i¾?= 1), and its approximation ratio is 2 i¾?logni¾?; the second one minimizes the combined cost f(i¾?) for 0 ≤ i¾?< 1, and its approximation ratio is $2 \sqrt{g}~\lceil \log n \rceil$.