Approximation algorithms for flexible graph connectivity

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 ≥ 1 , p ≥ 1 and q ≥ 0 be integers. In an instance of the ( p , q )-Flexible Graph Connectivity problem,...

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Vydané v:	Mathematical programming Ročník 204; číslo 1-2; s. 493 - 516
Hlavní autori:	Boyd, Sylvia, Cheriyan, Joseph, Haddadan, Arash, Ibrahimpur, Sharat
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Predmet:	Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical Secondary 90C17 Network design Edge-connectivity of graphs 05C40 Reliability of networks Approximation algorithms 90C59 90C27 Combinatorial optimization Primary 68W25
ISSN:	0025-5610, 1436-4646
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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 ≥ 1 , p ≥ 1 and q ≥ 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 S and a set of unsafe edges U , and nonnegative costs c : E → R ≥ 0 on the edges. A subset F ⊆ E of edges is feasible for the ( p , q ) -FGC problem if for any set F ′ ⊆ U with \| F ′ \| ≤ q , the subgraph ( V , F \ F ′ ) is p -edge 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 the ( 1 , 1 ) -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 ) -FGC problem extends to a ( k + 1 ) -approximation algorithm for the ( 1 , k ) -FGC problem. We present a 4-approximation algorithm for the ( k , 1 ) -FGC problem, and an O ( q log \| V \| ) -approximation algorithm for the ( p , q ) -FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted ( 1 , 1 ) -FGC problem by presenting a 16/11-approximation algorithm. The ( p , q ) -FGC problem is related to the well-known Capacitated k - Connected Subgraph problem (denoted Cap- k -ECSS ) that arises in the area of Capacitated Network Design. We give a min ( k , 2 u max ) -approximation algorithm for the Cap- k -ECSS problem, where u max denotes the maximum capacity of an edge.
AbstractList	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 ≥ 1 , p ≥ 1 and q ≥ 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 S and a set of unsafe edges U , and nonnegative costs c : E → R ≥ 0 on the edges. A subset F ⊆ E of edges is feasible for the ( p , q ) -FGC problem if for any set F ′ ⊆ U with \| F ′ \| ≤ q , the subgraph ( V , F \ F ′ ) is p -edge 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 the ( 1 , 1 ) -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 ) -FGC problem extends to a ( k + 1 ) -approximation algorithm for the ( 1 , k ) -FGC problem. We present a 4-approximation algorithm for the ( k , 1 ) -FGC problem, and an O ( q log \| V \| ) -approximation algorithm for the ( p , q ) -FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted ( 1 , 1 ) -FGC problem by presenting a 16/11-approximation algorithm. The ( p , q ) -FGC problem is related to the well-known Capacitated k - Connected Subgraph problem (denoted Cap- k -ECSS ) that arises in the area of Capacitated Network Design. We give a min ( k , 2 u max ) -approximation algorithm for the Cap- k -ECSS problem, where u max denotes the maximum capacity of an edge.
Author	Haddadan, Arash Cheriyan, Joseph Boyd, Sylvia Ibrahimpur, Sharat
Author_xml	– sequence: 1 givenname: Sylvia surname: Boyd fullname: Boyd, Sylvia organization: School of Electrical Engineering and Computer Science, University of Ottawa – sequence: 2 givenname: Joseph orcidid: 0000-0003-0316-7650 surname: Cheriyan fullname: Cheriyan, Joseph email: jcheriyan@uwaterloo.ca organization: Department of Combinatorics and Optimization, University of Waterloo – sequence: 3 givenname: Arash surname: Haddadan fullname: Haddadan, Arash organization: Modeling and Optimization, Amazon Inc – sequence: 4 givenname: Sharat orcidid: 0000-0002-1575-9648 surname: Ibrahimpur fullname: Ibrahimpur, Sharat organization: Department of Mathematics, London School of Economics and Political Science
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References_xml	– reference: Karger, D.R.: Global min-cuts in RNC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{R}}{\cal{N}}{\cal{C}}$$\end{document}, and other ramifications of a simple min-cut algorithm. In: Proceedings of the 4th Symposium on Discrete Algorithms, pp. 21–30 (1993) – reference: NagamochiHNishimuraKIbarakiTComputing all small cuts in an undirected networkSIAM J. Discrete Math.1997103469481145995210.1137/S08954801942713230884.05060 – reference: VaziraniVVApproximation Algorithms2003BerlinSpringer10.1007/978-3-662-04565-71005.68002 – reference: GabowHNGoemansMXTardosÉWilliamsonDPApproximating the smallest k-edge connected spanning subgraph by LP-roundingNetworks2009534345357253345410.1002/net.202891205.05125 – reference: GoemansMX WilliamsonDPThe Primal–Dual Method for Approximation Algorithms and Its Application to Network Design Problems, Chapter 41997BostonPWS Publishing Company1441911415.90101 – reference: FleischerLBuilding chain and cactus representations of all minimum cuts from Hao–Orlin in the same asymptotic run timeJ. Algorithms19993315172171269210.1006/jagm.1999.10390937.68154 – reference: WilliamsonDPGoemansMXMihailMVaziraniVVA primal–dual approximation algorithm for generalized Steiner network problemsCombinatorica1995153435454135728710.1007/BF012997470838.90133 – reference: ChakrabartyDChekuriCKhannaSKorulaNApproximability of capacitated network designAlgorithmica2015722493514334309910.1007/s00453-013-9862-41327.90023 – reference: Adjiashvili, D., Hommelsheim, F., Mühlenthaler, M.: Flexible graph connectivity. Math. Program. 1–33 (2021) – reference: GabowHNAn ear decomposition approach to approximating the smallest 3-edge connected spanning subgraph of a multigraphSIAM J. Discrete Math.20041814170211248810.1137/S08954801024054761071.05047 – reference: JainKA factor 2 approximation algorithm for the generalized Steiner network problemCombinatorica20012113960180571310.1007/s0049301700041107.68533 – reference: Adjiashvili, D., Hommelsheim, F., Mühlenthaler, M.: Flexible graph connectivity. In: Proceedings of the 21st Integer Programming and Combinatorial Optimization Conference, Volume 12125 of Lecture Notes in Computer Science, pp. 13–26 (2020) – reference: HunkenschröderCVempalaSVettaAA 4/3-approximation algorithm for the minimum 2-edge connected subgraph problemACM Trans. Algorithms2019154128402400110.1145/33415991454.68184 – reference: KhullerSVishkinUBiconnectivity approximations and graph carvingsJ. ACM1994412214235136920010.1145/174652.1746540822.68082 – reference: FrankAConservative weightings and ear-decompositions of graphsCombinatorica19931316581122117710.1007/BF012027900779.05033 – reference: MagnantiTLWongRTNetwork design and transportation planning: models and algorithmsTransp. Sci.198418115510.1287/trsc.18.1.10598.90038 – reference: GabowHNGallagherSIterated rounding algorithms for the smallest k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k}$$\end{document}-edge connected spanning subgraphSIAM J. Comput.201241161103288832210.1137/0807325721243.68225 – reference: Goemans, M.X., Goldberg, A.V., Plotkin, S.A., Shmoys, D.B., Tardos, É., Williamson, D.P.: Improved approximation algorithms for network design problems. 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Snippet	We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26,...
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SubjectTerms	Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical
Title	Approximation algorithms for flexible graph connectivity
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