Abstract
We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and Mühlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 133 (2021), IPCO 2020: pp. 1326). In an instance of the Flexible Graph Connectivity (FGC) problem, we have an undirected connected graph G = (V,E), a partition of E into a set of safe edges S and a set of unsafe edges U, and nonnegative costs {c_e}_{e ∈ E} on the edges. A subset F ⊆ E of edges is feasible for FGC if for any unsafe edge e ∈ F ∩ U, the subgraph (V,F⧵{e}) is connected. The algorithmic goal is to find a (feasible) solution F that minimizes c(F) = ∑_{e ∈ F} c_e. We present a simple 2approximation algorithm for FGC via a reduction to the minimumcost rout 2arborescence problem. This improves upon the 2.527approximation algorithm of Adjiashvili et al.
For integers p ≥ 1 and q ≥ 0, the (p,q)FGC problem is a generalization of FGC where we seek a minimumcost subgraph H = (V,F) that remains pedge connected against the failure of any set of at most q unsafe edges; that is, for any set F' ⊆ U with F' ≤ q, HF' = (V, F ⧵ F') should be pedge connected. Note that FGC corresponds to the (1,1)FGC problem. We give approximation algorithms for two important special cases of (p,q)FGC: (a) Our 2approximation algorithm for FGC extends to a (k+1)approximation algorithm for the (1,k)FGC problem. (b) We present a 4approximation algorithm for the (k,1)FGC problem.
For the unweighted FGC problem, where each edge has unit cost, we give a 16/11approximation algorithm. This improves on the result of Adjiashvili et al. for this problem.
The (p,q)FGC model with p = 1 or q ≤ 1 can be cast as the Capacitated kConnected Subgraph problem which is a special case of the wellknown Capacitated Network Design problem. We denote the former problem by CapkECSS. An instance of this problem consists of an undirected graph G = (V,E), nonnegative integer edgecapacities {u_e}_{e ∈ E}, nonnegative edgecosts {c_e}_{e ∈ E}, and a positive integer k. The goal is to find a minimumcost edgeset F ⊆ E such that every (nontrivial) cut of the capacitated subgraph H(V,F,u) has capacity at least k. We give a min(k, 2max_{e ∈ E} u_e)approximation algorithm for this problem.
BibTeX  Entry
@InProceedings{boyd_et_al:LIPIcs.FSTTCS.2021.9,
author = {Boyd, Sylvia and Cheriyan, Joseph and Haddadan, Arash and Ibrahimpur, Sharat},
title = {{Approximation Algorithms for Flexible Graph Connectivity}},
booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
pages = {9:19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772150},
ISSN = {18688969},
year = {2021},
volume = {213},
editor = {Boja\'{n}czy, Miko{\l}aj and Chekuri, Chandra},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15520},
URN = {urn:nbn:de:0030drops155206},
doi = {10.4230/LIPIcs.FSTTCS.2021.9},
annote = {Keywords: Approximation Algorithms, Combinatorial Optimization, Network Design, EdgeConnectivity of Graphs, Reliability of Networks}
}
Keywords: 

Approximation Algorithms, Combinatorial Optimization, Network Design, EdgeConnectivity of Graphs, Reliability of Networks 
Collection: 

41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) 
Issue Date: 

2021 
Date of publication: 

29.11.2021 