Extensions of the $(p,q)$-Flexible-Graph-Connectivity model

TL;DR

This paper develops approximation algorithms for the $(p,q)$-Flexible Graph Connectivity model, a network design problem involving safe and unsafe edges, ensuring $p$-edge connectivity despite removal of up to $q$ unsafe edges.

Contribution

It introduces new approximation algorithms for the generalized $(p,q)$-FGC model, extending previous work on flexible graph connectivity with improved solution methods.

Findings

Provided approximation algorithms for $(p,q)$-FGC.

Extended the model to more general network design scenarios.

Achieved bounds on the approximation ratios for the algorithms.

Abstract

We present approximation algorithms for network design problems in some models related to the $(p,q)$-FGC model. Adjiashvili, Hommelsheim and M\"uhlenthaler introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur introduced a generalization of FGC. Let $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges and a set of unsafe edges, and nonnegative costs $c\in\mathbb{R}_{\geq0}^E$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any set of unsafe edges, $F'$, with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible edge-set $F$ that minimizes $c(F) = \sum_{e \in F} c_e$.