Optimization algorithms for resilient path selection in networks

We study a Resilient Path Selection Problem (RPSP) arising in the design of communication networks with reliability guarantees. A graph is given, in which every arc has a cost and a probability of failure, and in which two nodes are marked as source and destination. The aim of our RPSP is to find a subgraph of minimum cost, containing a set of paths from the source to the destination nodes, such that the probability that all paths fail simultaneously is lower 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. In fact, we prove that even deciding if a given subgraph has a probability of failure not exceeding a given threshold is already NP-Complete. We therefore introduce an integer relaxation that simplifies the computation of such probability, and we design an exact algorithm for the full RPSP exploiting this relaxation and other ad hoc procedures. We present computational results, highlighting that our exact algorithms can handle graphs with up to 30 nodes within minutes of computing time, consistently producing proven optimal solutions. Moreover, we show that our algorithms can be used also as heuristics, outperforming path protection schemes from the literature also on much larger networks.