Approximation algorithms for flexible graph connectivity

Abstract

We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26, 2020, Math Program 1–33, 2021). Let $k\ge 1$, $p\ge 1$ and $q\ge 0$ be integers. In an instance of the (p, q)-Flexible Graph Connectivity problem, denoted $(p,q)\text {-FGC}$, we have an undirected connected graph $G = (V,E)$, a partition of E into a set of safe edges ${\mathscr {S}}$ and a set of unsafe edges ${\mathscr {U}}$, and nonnegative costs $c: E\rightarrow {\mathbb {R}}_{\ge 0}$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)\text {-FGC}$ problem if for any set $F'\subseteq {\mathscr {U}}$ with $|F'|\le q$, the subgraph $(V, F {\setminus } F')$ is p-edge connected. The algorithmic goal is to find a feasible solution F that minimizes $c(F) = \sum _{e \in F} c_e$. We present a simple 2-approximation algorithm for the $(1,1)\text {-FGC}$ problem via a reduction to the minimum-cost rooted 2-arborescence problem. This improves on the 2.527-approximation algorithm of Adjiashvili et al. Our 2-approximation algorithm for the $(1,1)\text {-FGC}$ problem extends to a $(k+1)$-approximation algorithm for the $(1,k)\text {-FGC}$ problem. We present a 4-approximation algorithm for the $(k,1)\text {-FGC}$ problem, and an $O(q\log |V|)$-approximation algorithm for the $(p,q)\text {-FGC}$ problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted $(1,1)\text {-FGC}$ problem by presenting a 16/11-approximation algorithm. The $(p,q)\text {-FGC}$ problem is related to the well-known Capacitated k-Connected Subgraph problem (denoted $\text {Cap-}k\text {-ECSS}$) that arises in the area of Capacitated Network Design. We give a $\min (k,2 {u}_{\textrm{max}})$-approximation algorithm for the $\text {Cap-}k\text {-ECSS}$ problem, where ${u}_{\textrm{max}}$ denotes the maximum capacity of an edge.