One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum $k$-Edge-Connected Spanning Subgraph problem as well as nonuniform demands such as the Survivable Network Design problem (SND). In a recent paper by [Dinitz, Koranteng, Kortsarz APPROX '22] , the authors observed that a weakness of these formulations is that it does not enable one to consider fault-tolerance in graphs that have just a few small cuts. To remedy this, they introduced new variants of these problems under the notion "relative" fault-tolerance. Informally, this requires not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. The problem is already highly non-trivial even for the case of a single demand. Due to difficulties introduced by this new notion of fault-tolerance, the results in [Dinitz, Koranteng, Kortsarz APPROX '22] are quite limited. For the Relative Survivable Network Design problem (RSND), when the demands are not uniform they give a nontrivial result only when there is a single demand with a connectivity requirement of $3$: a non-optimal $27/4$-approximation. We strengthen this result in two significant ways: We give a $2$-approximation for RSND where all requirements are at most $3$, and a $2^{O(k^2)}$-approximation for RSND with a single demand of arbitrary value $k$. To achieve these results, we first use the "cactus representation'' of minimum cuts to give a lossless reduction to normal SND. Second, we extend the techniques of [Dinitz, Koranteng, Kortsarz APPROX '22] to prove a generalized and more complex version of their structure theorem, which we then use to design a recursive approximation algorithm.

翻译：网络设计中最重要且研究最充分的场景之一是边连通性需求。这涵盖均匀需求（例如最小 $k$ 边连通生成子图问题）和非均匀需求（例如幸存网络设计问题）。在[Dinitz, Koranteng, Kortsarz APPROX '22]的近期论文中，作者指出这些公式的弱点在于无法处理仅有少量小割图的容错性问题。为此，他们引入了"相对"容错概念下的新变体。非正式地，这要求并非在有限故障数下两个节点保持连接（如经典场景），而是要求在有限故障数且故障后底层图中两节点连通时二者仍可连通。即使针对单一需求的情形，该问题也已高度非平凡。由于这种新型容错概念带来的困难，[Dinitz, Koranteng, Kortsarz APPROX '22]中的结果相当有限。对于相对幸存网络设计问题，当需求非均匀时，他们仅在单一需求且连通性要求为$3$时给出了非最优的$27/4$近似比。我们通过两个重要方向强化该结果：对需求值均不超过$3$的RSND给出$2$近似算法，并对单一任意需求值$k$的RSND给出$2^{O(k^2)}$近似算法。为获得这些结果，我们首先利用最小割的"仙人掌表示"实现到标准SND的无损归约，其次扩展[Dinitz, Koranteng, Kortsarz APPROX '22]的技术，证明其结构定理的广义化复杂版本，进而设计递归近似算法。