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.

我们研究具有可靠性保证的通信网络设计中出现的弹性路径选择问题（RPSP）。给出了一个图形，其中每个弧都有代价和故障概率，并且其中两个节点被标记为源和目标。我们的RPSP的目的是找到最小成本的子图，其中包含从源到目的节点的一组路径，以使所有路径同时发生故障的概率低于给定的阈值。我们探索了它的理论特性，并表明，尽管可以在多项式时间内解决一些有趣的特殊情况，但它通常是NP-hard。实际上，我们证明，即使确定给定的子图是否具有不超过给定阈值的失败概率也已经是NP-Complete。因此，我们引入了一个整数弛豫，它简化了这种概率的计算，并且我们利用此弛豫和其他临时程序设计了用于完整RPSP的精确算法。我们展示了计算结果，着重指出了我们精确的算法可以在几分钟内处理多达30个节点的图形，从而始终如一地提供经过验证的最佳解决方案。此外，我们证明了我们的算法也可以用作启发式算法，在更大的网络上也优于文献中的路径保护方案。持续生产经过验证的最佳解决方案。此外，我们证明了我们的算法也可以用作启发式算法，在更大的网络上也优于文献中的路径保护方案。持续生产经过验证的最佳解决方案。此外，我们证明了我们的算法也可以用作启发式算法，在更大的网络上也优于文献中的路径保护方案。