Survivable Network Design Revisited: Group-Connectivity

In the classical survivable network design problem (SNDP), we are given an undirected graph $G-(V,E)$ with costs on edges and a connectivity requirement $k(5,t)$ for each pair of vertices. The goal is to find a minimum-cost subgraph $H\sqsubseteq G$ such that every pair $(s,t)$ are connected by $k(s,t)$ edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an $O(k^{3}\log n)-$ approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This paper concerns the generalization of SNDP into the subset-to-subset setting, namely Group EC-SNDR We develop the framework, which yields the first non-trivial (true) approximation algorithm for Group. EC-SNDE Previously only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15l, and a true approximation algorithm is known only for the single-source variant with connectivity requirement $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor Comput. Sci’12].