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Given a number α with 0 α G = (V, E) and two nodes s and t in G, we consider the problem of finding two disjoint paths P1 and P2 from s to t such that length(P1) ≤ length(P2) and length(P1)+α·length(P2) is minimized. The paths may be node-disjoint or edge-disjoint, and the network may be directed or undirected. This problem has applications in reliable communication. We prove an approximation ratio ${1+\alpha} \over {2\alpha}$ for all four versions of this problem, and also show that this ratio cannot be improved for the two directed versions unless P = NP. We also present Integer Linear Programming formulations for all four versions of this problem. For a special case of this problem, we give a polynomial-time algorithm for finding optimal solutions.

Finding Two Disjoint Paths in a Network with Normalized α + -MIN-SUM Objective Function