Document

# Approximation Algorithms for Flexible Graph Connectivity

## File

LIPIcs.FSTTCS.2021.9.pdf
• Filesize: 0.74 MB
• 14 pages

## Acknowledgements

We thank the anonymous reviewers and PC members for their comments.

## Cite As

Sylvia Boyd, Joseph Cheriyan, Arash Haddadan, and Sharat Ibrahimpur. Approximation Algorithms for Flexible Graph Connectivity. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9

## 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. 1-33 (2021), IPCO 2020: pp. 13-26). 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 2-approximation algorithm for FGC via a reduction to the minimum-cost r-out 2-arborescence problem. This improves upon the 2.527-approximation 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 minimum-cost subgraph H = (V,F) that remains p-edge connected against the failure of any set of at most q unsafe edges; that is, for any set F' ⊆ U with |F'| ≤ q, H-F' = (V, F ⧵ F') should be p-edge 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 2-approximation algorithm for FGC extends to a (k+1)-approximation algorithm for the (1,k)-FGC problem. (b) We present a 4-approximation algorithm for the (k,1)-FGC problem. For the unweighted FGC problem, where each edge has unit cost, we give a 16/11-approximation 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 k-Connected Subgraph problem which is a special case of the well-known Capacitated Network Design problem. We denote the former problem by Cap-k-ECSS. An instance of this problem consists of an undirected graph G = (V,E), nonnegative integer edge-capacities {u_e}_{e ∈ E}, nonnegative edge-costs {c_e}_{e ∈ E}, and a positive integer k. The goal is to find a minimum-cost edge-set F ⊆ E such that every (non-trivial) 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
##### Keywords
• Approximation Algorithms
• Combinatorial Optimization
• Network Design
• Edge-Connectivity of Graphs
• Reliability of Networks

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. David Adjiashvili, Felix Hommelsheim, and Moritz Mühlenthaler. Flexible Graph Connectivity. In Proceedings of the 21st Integer Programming and Combinatorial Optimization Conference, volume 12125 of Lecture Notes in Computer Science, pages 13-26, 2020. URL: https://doi.org/10.1007/978-3-030-45771-6_2.
2. David Adjiashvili, Felix Hommelsheim, and Moritz Mühlenthaler. Flexible Graph Connectivity. Mathematical Programming, pages 1-33, 2021. URL: https://doi.org/10.1007/s10107-021-01664-9.
3. Deeparnab Chakrabarty, Chandra Chekuri, Sanjeev Khanna, and Nitish Korula. Approximability of Capacitated Network Design. Algorithmica, 72(2):493-514, 2015. URL: https://doi.org/10.1007/s00453-013-9862-4.
4. Lisa Fleischer. Building Chain and Cactus Representations of All Minimum Cuts from Hao-Orlin in the Same Asymptotic Run Time. Journal of Algorithms, 33(1):51-72, 1999. URL: https://doi.org/10.1006/jagm.1999.1039.
5. Harold N. Gabow, Michel X. Goemans, Éva Tardos, and David P. Williamson. Approximating the Smallest k-Edge Connected Spanning Subgraph by LP-rounding. Networks, 53(4):345-357, 2009. URL: https://doi.org/10.1002/net.20289.
6. Michel X. Goemans, Andrew V. Goldberg, Serge A. Plotkin, David B. Shmoys, Éva Tardos, and David P. Williamson. Improved Approximation Algorithms for Network Design Problems. In Proceedings of the 5th Symposium on Discrete Algorithms, pages 223-232, 1994. URL: https://doi.org/10.5555/314464.314497.
7. Michel X. Goemans and David P. Williamson. The Primal-Dual Method for Approximation Algorithms and its Application to Network Design Problems, chapter 4, pages 144-191. PWS Publishing Company, Boston, MA, 1997. URL: https://math.mit.edu/~goemans/PAPERS/book-ch4.pdf.
8. Kamal Jain. A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem. Combinatorica, 21(1):39-60, 2001. URL: https://doi.org/10.1007/s004930170004.
9. David R. Karger. Global Min-Cuts in ℛ𝒩𝒞, and Other Ramifications of a Simple Min-Cut Algorithm. In Proceedings of the 4th Symposium on Discrete Algorithms, pages 21-30, 1993. URL: https://doi.org/10.5555/313559.313605.
10. Samir Khuller and Uzi Vishkin. Biconnectivity Approximations and Graph Carvings. Journal of the ACM, 41(2):214-235, 1994. URL: https://doi.org/10.1145/174652.174654.
11. Thomas L. Magnanti and Richard T. Wong. Network Design and Transportation Planning: Models and Algorithms. Transportation Science, 18(1):1-55, 1984. URL: https://doi.org/10.1287/trsc.18.1.1.
12. Polina Rozenshtein, Aristides Gionis, B. Aditya Prakash, and Jilles Vreeken. Reconstructing an Epidemic Over Time. In Proceedings of the 22nd International Conference on Knowledge Discovery and Data Mining, KDD '16, pages 1835-1844, 2016. URL: https://doi.org/10.1145/2939672.2939865.
13. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer-Verlag Berlin Heidelberg, 2003.
14. András Sebö and Jens Vygen. Shorter Tours by Nicer Ears: 7/5-Approximation for the Graph-TSP, 3/2 for the Path Version, and 4/3 for Two-Edge-Connected Subgraphs. Combinatorica, 34(5):597-629, 2014. URL: https://doi.org/10.1007/s00493-014-2960-3.
15. Lawrence V. Snyder, Maria P. Scaparra, Mark S. Daskin, and Richard L. Church. Planning for Disruptions in Supply Chain Networks, pages 234-257. Institute for Operations Research and the Management Sciences (INFORMS), 2014. URL: https://doi.org/10.1287/educ.1063.0025.
16. Vijay V. Vazirani. Approximation Algorithms. Springer Berlin Heidelberg, 2003. URL: https://doi.org/10.1007/978-3-662-04565-7.
17. David P. Williamson, Michel X. Goemans, Milena Mihail, and Vijay V. Vazirani. A Primal-Dual Approximation Algorithm for Generalized Steiner Network Problems. Combinatorica, 15(3):435-454, 1995. URL: https://doi.org/10.1007/BF01299747.