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20 April K. Andreas and J. Cole Smith2s n10.9s =1 2ns n0.80.70.60.50.40.30.2( s ˆ1 , sˆ2)nIn⎛⎜ sˆ⎝1nII− ˆ,τ s1−sˆ1n1n⎟ ⎞⎠0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11s nFigure 1.Illustration of First Iteration of Vertical Branch-and-Bound Algorithmous ways that the interval could be split to prevent the nonlinear-infeasible solution from recurring(e.g., Figure 1 shows a “vertical branching" rule). We examine the advantages and disadvantagesof these methods and present computational results to support the use of one particularmethod; specifically, choosing a partitioning for which the distance from the nonlinearinfeasiblesolution to either partitioning plane is equal, thereby maximizing the minimum distanceto either partitioned region. This comparatively effective method of branching does notrely on the axes to generate the cuts, and generalizes the Reformulation-Linearization Technique[8] (for continuous-variables optimization problems) on this particular problem.Relaxing the edge-disjoint constraints will necessitate adjustments to be made to RTP-D.The formulation and algorithm adjustments to solve the shared-arcs variation of our problem(RTP-S) must then be modified. Our approach now requires the introduction of an additionaldecision variable and moves our linear chord relaxation into a three-dimensional space. (Detailsare omitted in this paper due to space limitations.) Model tightening via pruning and lifting,as well as preprocessing is also less straightforward for RTP-S, though it is still possible. Aspecialized partitioning procedure proves difficult, however, for this particular problem dueto the fact that joint reliability constraint is now neither convex nor concave. We work aroundthis problem by implementing the Reformulation-Linearization Technique [8] to assist in partitioningour space for the branch-and-bound procedure. We provide computational resultsto recommend the best pruning, lifting, and partitioning strategies for solving RTP-S.A natural extension to the two-disjoint-path problem is the k-disjoint-path problem fork > 2. The previous approach would suffer due to the increasing difficulties of relaxingmultiple nonlinear terms as k increases. An alternative approach is to consider a path-basedformulation, in which variables x p equal to one if path p ∈ P is selected, and zero otherwise,where P is the entire set of origin-destination paths in the network. However, there are toomany such paths to generate, and a column generation approach would have to deal with