Mathematical Formulations for the Optimal Design of Resilient Shortest Paths
We study a Resilient Shortest Path Problem (RSPP) arising in the literature for the design of communication networks with reliability guarantees. A graph is given, in which every edge has a cost and a probability of availability, and in which two vertices are marked as source and destination. The aim of our RSPP is to find a subgraph of minimum cost, containing a set of paths from the source to the destination vertices, such that the probability that at least one path is available is higher than a given threshold. We explore its theoretical properties and show that, despite a few interesting special cases can be solved in polynomial time, it is in general NP-hard. Computing the probability of availability of a given subgraph is already NP-hard; we therefore introduce an integer relaxation that simplifies the computation of such probability, and we design a corresponding exact algorithm. We present computational results, finding that our algorithm can handle graphs with up to 20 vertices within minutes of computing time.
KeywordsNetwork reliability Shortest path problem Branch and price Column generation
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