Compact Models for Hop-Constrained Node Survivable Network Design: An Application to MPLS

Design of survivable networks with vulnerability constraints

“…We will consider two building blocks, one corresponding to the design of the primary path, and the other to the set of the backup edges. While models for designing the primary path can be straightforwardly taken from the literature (see, e.g., [5,6] and [17] for a more recent article), this is not true for models used to determine the optimal set of backup edges. In Section 3.2, we will provide three characterizations of valid sets of backup edges.…”

“…Relating the LP bounds of the two formulations We start by pointing out that in several situations, the aggregated model produces the same LP bound as the disaggregated model, see for instance [6], in which case it is clearly preferable to the disaggregated one. In fact, it is also with such an aggregated model that the good results reported in Botton et al [3] are obtained.…”

Hop-Constrained Node Survivable Network Design: An Application to MPLS over WDM

“…Following Gouveia et al (2006) we present next a hop-indexed formulation for the problem, which involves three sets of variables. Let N i (i ∈ V \ S) be binary variables indicating whether a core LSR is put in operation on node i, let u e (e ∈ E ) be integer variables representing the number of lightpaths installed on edge e, and let w hpq ij ( p, q ∈ S; {i, j} ∈ E ; h = 1, ..., H)(the variables with i = q and j = q are not defined and, also, with j = p) be binary variables indicating whether a ( p, q) − H − path traverses edge {i, j} in the direction from i to j and in position h. Note that, as explained in Gouveia et al (2003Gouveia et al ( , 2006, when i = j = q we have "loop" variables w hpq qq (h = 2, ..., H) which model situations when a path (or paths) from node p to node q contains fewer than H arcs (that is, w hpq jq = 1 for some j ∈ V \ {q} and 1 ≤ h ≤ H − 1).…”

“…The other formulation is an aggregated version of the previous one and, for each pair of demand nodes, uses a single hop-indexed flow system for the two paths. The main result of Gouveia et al (2006) states that the linear programming relaxation of the two formulations are equivalent. That is, in this case, aggregation does not lead to a formulation with a weaker linear programming relaxation.…”

“…Their conclusions were that these hop-indexed formulations are rather efficient in designing networks with delay sensitive services, and, for larger problem instances, they also have devised a heuristic procedure based on decomposing the complete model into two simpler ILP models solved sequentially (which can be easily adapted to our more general problem). Two compact hop-indexed ILP formulations for the problem of designing a single network layer (MPLS) with survivability based on two (D = 2) node disjoint paths between any pair of ingress/egress nodes are discussed in Gouveia et al (2006). One of the formulations uses a hop-indexed path system for each single path.…”