Mathematical Programming Algorithms for Two-Path Routing Problems with Reliability Considerations

Most traditional routing problems assume perfect operability of all arcs and nodes. However, when independent arc failure probabilities exist, a secondary objective must be present to retain some measure of expected functionality, introducing nonlinear, nonconvex constraints. We examine the Robust Two-Path Problem, which seeks to establish two paths between a source and destination node wherein at least one path must remain fully operable with some threshold probability. We consider the case where both paths must be arc-disjoint and the case where arcs can be shared between the paths. We begin by proving the NP-hardness of these problems, and then examine various strategies for solving the resulting nonlinear integer program, including pruning, coefficient tightening, lifting, and branch-and-bound partitioning schemes. We discuss the advantages and disadvantages of these methods, and conclude with computational results. Subject Classifications: Programming, Integer: Nonlinear. Networks, Theory. Programming, Integer: Branch-and-bound Area of Review: Optimization 1 Problem Framework In this paper, we examine a class of routing problems on a directed graph G(N, A), where N is the set of nodes {1, ..., n} and A is the set of arcs. Each arc (i, j) ∈ A has a usage cost cij and a reliability 0 < pij ≤ 1, which denotes the probability that arc (i, j) successfully operates. These probabilities are assumed to be independent. The fundamental problem that we consider in this paper seeks to find the minimum-cost pair of directed paths from an origin node to a destination node, such that the probability that all arcs successfully operate in at least one path is larger than some threshold value τ , where 0 < τ < 1. In one variation of this problem, we enforce a restriction that the paths must be arc-disjoint, while in the other variation, arcs may be shared by both paths. Two paths are said to be arc-disjoint if no arc belongs to both of these paths. Note that in the latter case, the failure of a shared arc causes both paths to fail. Consider the example in Figure 1. Each of the thirteen arcs is labeled with its associated cost and reliability. For this example, suppose τ = 0.65. If we require that the two paths must be arc-disjoint, then the optimal solution uses paths ABFH and ADGH at a cost of 31, and a joint reliability of 72.6%. If we consider the case in which the two paths can share arcs, then the best solution uses paths ABEH and ABFH, at a total cost of only 27 and a joint reliability of 67%. As an introduction to our problem, consider the single shortest-path problem that minimizes the sum of arc usage costs, subject to the side constraint that the successful traversal of the path is greater than some threshold value τ . For 0 < τ < 1, it is not hard to show that this problem is ordinarily NP-hard. (By contrast, we will demonstrate that the two-path problem considered in this paper is strongly NP-hard for 0 < τ < 1.) The reliability-constrained shortest-path problem is related to several other single-path problems that have appeared in the literature. This problem falls into the category of a shortest-path problem with an additional linear constraint, since the reliability condition can be converted to a linear constraint by a simple use of logarithms. Such problems are referred to as resource-constrained shortest-path problems in the literature, since the side constraints on these problems can represent limitations on the consumption of resources as the path is traversed. Joksch (1966) presented both a linear programming and a dynamic programming approach for solving the shortest-path problem subject to a side constraint for a graph with nonnegative costs and constraints. One type of method used to solve this problem applies approximation schemes, such as those introduced by Hassin (1992), to a reduced graph. Another method is to enumerate paths (the k-shortest path problem) in order of increasing reduced cost and terminate when the constraint is met. This is suggested by Handler and Zang (1980), who developed a dual algorithm for the constrained shortest-path problem, developed further by Beasley and Christofides (1989). Mehlhorn and Ziegelmann (2000) suggest a method of enumerating paths in order of increasing reduced cost combined with on-line pruning, and also show the resource-constrained shortest-path problem to be NP-complete.