Survivable Network Design with Group-to-Group Requirement

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 ( s , t ) for each pair of vertices. The goal is to find a minimum-cost subgraph H ⊆ G such that every pair ( s , t ) is 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 EC-SNDP into the subset-to-subset setting, namely Group EC-SNDP. We develop a framework, which yields the first non-trivial (true) approximation algorithm for Group EC-SNDP. Previously, only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15], and a true approximation algorithm is known only for the single-source variant with connectivity requirement k ( S , T ) ∈ {0, 1, 2} [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor. Comput. Sci.’12]. On the negative side, in terms of the number of connectivity demands q , we give an Ω ( q /log q )-hardness result for large k , complementing the previous inapproximability results, e.g., hardness in terms of k : k 1/5 − ε -hardness [Cheriyan et al., SODA’12; Laekhanukit, SODA’14; Chalermsook et al., SODA’15; Manurangsi, IPL’19]; hardness in terms of n : \(2^{\log ^{1-\varepsilon }n} \) -hardness [Chalermsook et al., SODA’15].