By Christelle Wynants (auth.)

As the telecommunication introduces new refined applied sciences, the character of providers and the quantity of calls for have replaced. certainly, a huge variety of recent prone for clients seem, combining voice, info, pictures, video, and so forth. this means new making plans matters. Fiber transmission structures that could hold quite a lot of information on a couple of strands of twine have been brought. those platforms have the sort of huge bandwidth that the failure of even a unmarried transmission hyperlink: within the community can create a critical provider loss to buyers. as a result, a truly excessive point of carrier reliability is changing into central for either approach clients and repair services. due to the fact that apparatus mess ups and injuries can't be shunned solely, networks need to be designed so one can "survive" mess ups. this is often performed via judiciously fitting spare means over the community in order that all site visitors interrupted via a failure should be diverted round that failure in terms of this spare or reserve capability. This after all interprets into large investments for community operators. Designing such survivable networks whereas minimizing spare capability expenses is, no longer strangely, a huge challenge of working businesses which provides upward push to very tough combinatorial difficulties. in an effort to make telecommunication networks survivable, you could primarily use varied thoughts: safeguard or recovery. the security strategy preas symptoms spare means to guard every one portion of the community independently, whereas the recovery strategy spreads the redundant potential over the entire community and makes use of it as required so as to fix the disrupted traffic.

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1: (a) requirement graph (b) feasible solution. 5. 4. 4. 2: requirement graph. A feasible solution of this FULR(I) instance is given by Ye = 1/4 VeE E for a total cost of 15/4. If we solve the FULR( 1) problem on the two connected components, the optimal solutions are Yab = Yac = Ybc = 1 (cost = 3) and Yde = Ydj = Yje = 1 (cost = 3) Single commodity flow requirements and the sum of the costs is 6> 15/4. 47 0 We now want to construct an interval for the optimal solution value of any FULR(1) instance.

They propose a polynomial algorithm easy to implement to obtain an optimal solution. They extend their work in [33] to consider the case of multicommodity flow requirements and capacity costs given by a linear function of edge length. e. constraint generation) to solve the linear relaxation of the problem (relaxing the capacity variables). Their larger instance has 10 nodes and 20 edges. Still in the sixties, the same problem is studied by Chien [12] who uses a partitioning approach. The special multi-hour problem introduced by Gomory and Hu [32], is again approached by Talluri [79].

2n = 1 -+ 0 2n(n - 2) (n - 2) when n -+ 00. • In summary, when the requirement graph C r is complete, an optimal solution graph of the FULR(1) problem is the complete graph with edge capacities equal to (l/(n - 2)) and this solution is asymptotically optimal for any requirement graph. However, if r contains a Hamiltonian cycle, then an optimal solution of the FULR(l) problem is obtained by installing half a unit of capacity on the edges of this cycle. f. Dirac [17]). a a Single commodity flow requirements 51 The second part of this section is dedicated to the class of FULR(l) problems with complete multipartite requirement graphs.