在非均匀边故障下保护图的连通性
Protecting the Connectivity of a Graph Under Non-Uniform Edge Failures
AI总结：
本文提出非均匀边故障下的 $(p,q)$-Steiner-Connectivity Preservation 问题，给出小参数精确算法、一般情形近似算法，并证明输入相关情形的 NP 完全性。
摘要：
我们研究通过保护或强化边来保证给定图连通性的问题。在此，受保护的边被假定为鲁棒的且不会发生故障，这体现了一种非均匀故障模型。我们引入 $(p,q)$-Steiner-Connectivity Preservation 问题：在假定至多有 $q$ 条未受保护的边可能发生故障的情况下，保护一个最小成本的边集合，使底层图在给定终端对之间面对边故障时仍保持 $p$-边连通性。我们针对 $p$ 和 $q$ 较小的情形设计了多项式时间精确算法，并针对 $p$ 和 $q$ 的一般取值设计了近似算法。此外，我们表明，当 $p$ 和 $q$ 都作为输入的一部分时，即使判定给定解是否可行也是 NP 完全的。这一困难性也延伸到 Flexible Network Design，这是一个已受到广泛关注的研究方向。特别地，以往工作聚焦于 $p$ 或 $q$ 之一为常数的问题设定，而我们的新困难性结果现在为这些设定提供了依据。
英文摘要：
We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a non-uniform failure model. We introduce the $(p,q)$-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains $p$-edge-connectivity between given terminal pairs against edge failures, assuming at most $q$ unprotected edges can fail. We design polynomial-time exact algorithms for the cases where $p$ and $q$ are small and approximation algorithms for general values of $p$ and $q$. Additionally, we show that when both $p$ and $q$ are part of the input, even deciding whether a given solution is feasible is NP-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either $p$ or $q$ is constant, for which our new hardness result now provides justification.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
