Approximation Algorithms for Relative Survivable Network Design Problems

Abstract

The Survivable Network Design (SND) problem is a classical and well-studied graph connectivity problem. Given a set of source-sink pairs and demands between them, SND asks one to compute a subgraph such that the number of paths between each pair meets their demand. SND is primarily interesting in modeling fault-tolerance; we can see the problem as requiring certain nodes to be connected even if some edges ”fail”. It is well known that a 2-approximation algorithm for SND exists, using the method of iterative rounding. In 2022, Dinitz et al. introduced a problem that we refer to as Path-Relative Survivable Network Design (PRSND), a natural extension of SND that addresses cases where the underlying graph does not have the required connectivity; in this problem, we require that the connectivity of our subgraph is ”as good as” it is in the original graph. Perhaps surprisingly, this variation makes PRSND much harder to approximate than standard SND, and outside of certain special cases no constant-factor approximations have been found. In this thesis we introduce the Cut-Relative Survivable Network Design (CRSND) prob- lem, another variant of SND that similarly aims to capture relative fault-tolerance. We show that this problem admits a 2-approximation algorithm, matching the best known ap- proximation factor for SND, via a decomposition technique. We explore some properties of said approximation, as well as hardness and modeling properties of Cut-Relative Network Design problems.