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Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks

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Abstract

We consider the problems of minimum-cost design and augmentation of directed network clusters that have diameter 2 and maintain the same diameter after the deletion of up to R elements (nodes or arcs) anywhere in the cluster. The property of a network to maintain not only the overall connectivity, but also the same diameter after the deletion of multiple nodes/arcs is referred to as strong attack tolerance. This paper presents the proof of NP-completeness of the decision version of the problem, derives tight theoretical bounds, as well as develops a heuristic algorithm for the considered problems, which are extremely challenging to solve to optimality even for small networks. Computational experiments suggest that the proposed heuristic algorithm does identify high-quality near-optimal solutions; moreover, in the special case of undirected networks with identical arc construction costs, the algorithm provably produces an exact optimal solution to strongly attack-tolerant two-hop network design problem, regardless of the network size.

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Notes

  1. The parameter β of the generated power-law graphs was chosen to approximately match the number of arcs in the corresponding uniform graphs with the same number of nodes. This procedure was performed for comparison reasons. It is described in more detail in Veremyev and Boginski (2012a).

References

  • Balasundaram B, Butenko S, Trukhanov S (2005) Novel approaches for analyzing biological networks. J Comb Optim 10:23–39

    Article  MATH  MathSciNet  Google Scholar 

  • Bang-Jensen J, Chiarandini M, Morling P (2010) A computational investigation of heuristic algorithms for 2-edge-connectivity augmentation. Networks 55(4):299–325

    MATH  MathSciNet  Google Scholar 

  • Bang-Jensen J, Jordán T (1998) Edge-connectivity augmentation preserving simplicity. SIAM J Discrete Math 11(4):603–623

    Article  MATH  MathSciNet  Google Scholar 

  • Bendali F, Diarrassouba I, Mahjoub AR, Mailfert J (2010) The k edge-disjoint 3-hop-constrained paths polytope. Discrete Optim 7(4):222–233

    Article  MATH  MathSciNet  Google Scholar 

  • Botton Q, Fortz B, Gouveia L, Poss M (2011) Benders decomposition for the hop-constrained survivable network design problem. INFORMS J Comput. doi:10.1287/ijoc.1110.0472

    Google Scholar 

  • Conforti M, Galluccio A, Proietti G (2004) Edge-connectivity augmentation and network matrices. In: Hromkovic J, Nagl M, Westfechtel B (eds) WG. Lecture notes in computer science, vol 3353. Springer, Berlin, pp 355–364

    Google Scholar 

  • IBM ILOG CPLEX optimization studio 12.4 (2011) http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/

  • Dahl G, Gouveia L (2004) On the directed hop-constrained shortest path problem. Oper Res Lett 32(1):15–22

    Article  MATH  MathSciNet  Google Scholar 

  • Dahl G, Huygens D, Mahjoub AR, Pesneau P (2006) On the edge-disjoint 2-hop-constrained paths polytope. Oper Res Lett 34(5):577–582

    Article  MATH  MathSciNet  Google Scholar 

  • Dahl G, Johannessen B (2004) The 2-path network problem. Networks 43(3):190–199

    Article  MATH  MathSciNet  Google Scholar 

  • Eswaran KP, Tarjan RE (1976) Augmentation problems. SIAM J Comput 5(4):653–665

    Article  MATH  MathSciNet  Google Scholar 

  • Fico™ xpress optimization suite 7.1 (2011) http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx

  • Frank A (1992) Augmenting graphs to meet edge-connectivity requirements. SIAM J Discrete Math 5(1):25–53

    Article  MATH  MathSciNet  Google Scholar 

  • Frank A (1994) Connectivity augmentation problems in network design. In: Birge J, Murty KG (eds) Mathematical programming: state of the art. University of Michigan Press, Ann Arbor, pp 34–63

    Google Scholar 

  • Fured Z, Horak P, Pareek CM, Zhu X (1998) Minimal oriented graphs of diameter 2. Graphs Comb 14:345–350

    Article  Google Scholar 

  • Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York

    MATH  Google Scholar 

  • Gouveia L, Patricio P, Sousa A (2008) Hop-constrained node survivable network design: an application to mpls over wdm. Netw Spat Econ 8(1):3–21

    Article  MATH  MathSciNet  Google Scholar 

  • Hsu TS (2002) Simpler and faster biconnectivity augmentation. J Algorithms 45(1):55–71

    Article  MATH  MathSciNet  Google Scholar 

  • Huygens D, Labbé M, Mahjoub AR, Pesneau P (2007) The two-edge connected hop-constrained network design problem: valid inequalities and branch-and-cut. Networks 49(1):116–133

    Article  MATH  MathSciNet  Google Scholar 

  • Huygens D, Mahjoub AR, Pesneau P (2004) Two edge-disjoint hop-constrained paths and polyhedra. SIAM J Discrete Math 18(2):287–312

    Article  MATH  MathSciNet  Google Scholar 

  • Ishii T, Nagamochi H (2000) On the minimum augmentation of an l-connected graph to a k-connected graph. In: SWAT, pp 286–299

    Google Scholar 

  • Li CL, McCormick S, Simchi-Levi D (1992) On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems. Oper Res Lett 11:303–308

    Article  MATH  MathSciNet  Google Scholar 

  • Ljubić I (2010) A branch-and-cut-and-price algorithm for vertex-biconnectivity augmentation. Networks 56(3):169–182

    Article  MATH  MathSciNet  Google Scholar 

  • Meijer H, Dawes R (1988) Fault tolerant networks of specified diameter. In: van Leeuwen J (ed) WG. Lecture notes in computer science, vol 344. Springer, Berlin, pp 74–86

    Google Scholar 

  • Mokken RJ (1979) Cliques, clubs and clans. Qual Quant 13(2):161–173

    Article  Google Scholar 

  • Murty USR (1968) On critical graphs of diameter 2. Math Mag, 138–140

  • Pirkul H, Soni S (2003) New formulations and solution procedures for the hop constrained network design problem. Eur J Oper Res 148(1):126–140

    Article  MATH  Google Scholar 

  • Schumacher U (1984) An algorithm for construction of a k-connected graph with minimum number of edges and quasiminimal diameter. Networks 14:63–74

    Article  MATH  MathSciNet  Google Scholar 

  • Soneoka T, Nakada H, Imase M (1990) Design of a d-connected digraph with a minimum number of edges and a quasiminimal diameter. Discrete Appl Math 27:255–265

    Article  MATH  MathSciNet  Google Scholar 

  • Veremyev A, Boginski V (2012a) Identifying large robust network clusters via new compact formulations of maximum k-club problems. Eur J Oper Res 218:316–326

    Article  MATH  MathSciNet  Google Scholar 

  • Veremyev A, Boginski V (2012b) Robustness and strong attack tolerance of low-diameter networks. In: Sorokin A, Murphey R, Thai MT, Pardalos PM (eds) Dynamics of information systems: mathematical foundations. Springer, Berlin (to appear)

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Authors and Affiliations

  1. Industrial and Systems Engineering Department, University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    Grigory Pastukhov, Alexander Veremyev, Vladimir Boginski & Eduardo L. Pasiliao

  2. Air Force Research Laboratory, Munitions Directorate, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

Authors
  1. Grigory Pastukhov
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  2. Alexander Veremyev
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  3. Vladimir Boginski
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  4. Eduardo L. Pasiliao
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Corresponding author

Correspondence to Vladimir Boginski.

Additional information

This work was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA).

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Pastukhov, G., Veremyev, A., Boginski, V. et al. Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks. J Comb Optim 27, 462–486 (2014). https://doi.org/10.1007/s10878-012-9523-6

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