Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

Abstract. Let G = (N,A) be an undirected graph with n nodes and m arcs, a designated source node s and a sink node t. This paper addresses sensitivity analysis questions concerning the shortest s-t path (SP) problem in G and the maximum capacity s-t path (MCP) problem in G. Suppose that P* is a shortest s-t path in G with respect to a nonnegative distance vector c. For each arc e ∈ A, the lower SP tolerance of an arc e is the minimum non-negative value that the length of arc e can take (with all other lengths staying fixed) so that P* remains an optimal path. Similarly, the upper SP tolerance of an arc e is the maximum value that the length of arc e can take so that P* remains an optimal path. We show that the problem of finding all upper and lower tolerances of arcs in A can be solved in O(min(n, m log n)) time. Moreover, the problem of finding all tolerances is computationally equivalent to the "Minimum Cost Interval Problem" which we describe as follows. For each i = 1 to m, let [ai, bi] denote an interval with endpoints in {1, ..., n}, and an associated cost ci. For each k = 1 to n, identify a minimum cost interval [ai, bi] containing k.