The Complexity of Traffic Grooming in Optical Path Networks with Egress Traffic

Abstract

We consider the problem of minimizing network costs when grooming traffic in optical networks that support Wavelength Division Multiplexing (WDM). While the general problem has been shown to be NP-Hard for a number of cost measures, there still exist restricted problems for which no complexity bound is known. In this research, we restrict our attention to traffic grooming for path networks with egress (all-to-one) traffic. This restricted model has practical significance for high speed (optical) access networks and can also lead to better bounds and approximations on more general network topologies (such as ring and star networks) that can be decomposed into path networks. Three important cost measures for this restricted model are studied. The first cost measure is the total number of ADMs used by the solution. Minimizing this cost was known to be NP-Complete even for egress traffic without using cross connects. We show that allowing an unbounded number of wavelengths obviates the need for digital cross connects at the nodes and hence the problem remains NP-Complete even when cross connects are allowed. The second cost measure is the number of transceivers used by the solution. We show that the problem of minimizing the number of transceivers is NP-Complete, even when restricted to egress traffic. We then develop a simple approximation scheme where the transceiver cost exceeds the minimum by at most the number of required wavelengths. Finally, we show that under certain conditions, there exist solutions that simultaneously minimize both ADM and transceiver costs. The third cost model aims to minimize the total electronic switching in the network. For this cost measure, we develop a polynomial time algorithm to determine the cost and structure of an optimum solution when the wavelength capacity constraint is relaxed. A closed form expression to determine the minimum cost is presented for problem instances with uniform traffic. We observe that these costs provide a lower bound on the cost of solutions to problems with finite capacity. In addition, the structure of the solution for infinite capacity wavelengths is used to obtain an upper bound for instances with finite capacity having uniform unit traffic. It is already known that the problem of minimizing this cost is NP-Complete for path networks with any-to-any traffic, even when a virtual topology is already specified. We show that for networks with egress traffic, given a virtual topology, there do exist polynomial time algorithms for minimizing this cost. Finally, we present an algorithm to minimize the cost when the number of wavelengths is fixed at two.