BACKGROUND Solving the shortest path and min-cut problems are key in achieving high-performance and robust communication networks. Those problems have often been studied in deterministic and uncorrelated networks both in their original formulations as well as in several constrained variants. However, in real-world networks, link weights (e.g., delay, bandwidth, failure probability) are often correlated due to spatial or temporal reasons, and these correlated link weights together behave in a different manner and are not always additive, as commonly assumed. METHODS In this paper, we first propose two correlated link weight models, namely (1) the deterministic correlated model and (2) the (log-concave) stochastic correlated model. Subsequently, we study the shortest path problem and the min-cut problem under these two correlated models. RESULTS AND CONCLUSIONS We prove that these two problems are NP-hard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time. However, these two problems are solvable in polynomial time under the (constrained) nodal deterministic correlated model, and can be solved by convex optimization under the (log-concave) stochastic correlated model.

中文翻译：

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相关网络中的优化问题。

背景技术解决最短路径和最小切问题是实现高性能和鲁棒通信网络的关键。这些问题经常在确定性和不相关的网络中进行研究，无论是在其原始公式中还是在几个受约束的变体中。然而，在现实世界的网络中，链路权重（例如，延迟、带宽、故障概率）通常由于空间或时间原因而相关，并且这些相关的链路权重一起以不同的方式表现并且并不总是像通常假设的那样总是相加的. 方法在本文中，我们首先提出了两个相关的链接权重模型，即（1）确定性相关模型和（2）（对数凹）随机相关模型。随后，我们研究了这两个相关模型下的最短路径问题和最小割问题。结果与结论我们证明了这两个问题在确定性相关模型下是NP-hard问题，甚至不能在多项式时间内逼近到任意程度。然而，这两个问题在（受约束的）节点确定性相关模型下可以在多项式时间内解决，并且可以通过（对数-凹）随机相关模型下的凸优化来解决。