Improved Approximations for Relative Survivable Network Design

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 one small cut. 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.