Design of edge-disjoint path protected WDM networks: asymptotic optimality of shortest path

We address the problem of allocating fibers (each supporting only a limited set of wavelengths) on the links of a WDM network at minimum cost, such that a set of edge-disjoint path protected connection requests can be realized. The cost of a link is assumed to be linear in the number of fibers rather than being linear in the number of wavelengths used on this link, reflecting modular capacity considerations. Therefore, a solution induced by routing each connection "simply" along the minimum-cost (shortest) pair of edge-disjoint lightpaths may not minimize the total fiber cost. In this paper we quantify the increase in the total fiber cost due to this simple routing strategy. In particular, we prove that the cost of a solution induced by routing along shortest path pairs is guaranteed to lie within a certain factor of the minimum possible cost. This leads also to the fact that the cost of this solution is asymptotically minimum in heavily loaded networks, and in networks that are large, sparse and supporting all-to-all communications. En route, we prove that the optimal objective function value of the linear programming (LP) relaxation actually corresponds to routing along shortest path pairs. We have thus presented a provably good upper bound and a lower bound on the total fiber cost, that can be computed in polynomial-time, and can be used as benchmarks against which exact and heuristic approaches are compared.