Abstract
Hassin et al. [9] consider the Max-Exp-Cover-R problem to study the facility location problem on a graph in the presence of unreliable links when the link failure is according to the Linear Reliability Order (LRO) model. They showed that for unbounded R the problem is polynomial time solvable and for \(R=1\) and planar graphs the problem is NP-Complete. In this paper, we study the Max-Exp-Cover-1 problem under the LRO edge failure model. We obtain a fixed parameter tractable algorithm for Max-Exp-Cover-1 problem for bounded treewidth graphs, parameterized by the treewidth. We extend the Baker’s technique (Baker, J. ACM 1994) to obtain PTAS for Max-Exp-Cover-1 problem under the LRO model on planar graphs. We observe that the coverage function of the Max-Exp-Cover-R problem is submodular and the problem admits a \((1-1/e)\)-approximation for any failure model in which the expected coverage of a set by another set can be computed in polynomial time.
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References
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)
Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1), 93–109 (1990)
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–23 (1993)
Colbourn, C.J., Xue, G.: A linear time algorithm for computing the most reliable source on a series-parallel graph with unreliable edges. Theor. Comput. Sci. 209(1), 331–345 (1998)
Daskin, M.S.: A maximum expected covering location model: formulation, properties and heuristic solution. Transp. Sci. 17(1), 48–70 (1983)
Ding, W.: Computing the most reliable source on stochastic ring networks. In: 2009 WRI World Congress on Software Engineering, vol. 1, pp. 345–347, May 2009
Ding, W.: Extended most reliable source on an unreliable general network. In: 2011 International Conference on Internet Computing and Information Services, pp. 529–533, September 2011
Ding, W., Xue, G.: A linear time algorithm for computing a most reliable source on a tree network with faulty nodes. Theor. Comput. Sci. 412(3), 225–232 (2011). Combinatorial Optimization and Applications
Hassin, R., Ravi, R., Salman, F.S.: Tractable cases of facility location on a network with a linear reliability order of links. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 275–276. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04128-0_24
Hassin, R., Ravi, R., Salman, F.S.: Multiple facility location on a network with linear reliability order of edges. J. Comb. Optim. 34(3), 1–25 (2017)
Kloks, T. (ed.): Treewidth: Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0045375
Lovasz, L.: Matching Theory (North-Holland Mathematics Studies). Elsevier Science Ltd., Oxford (1986)
Melachrinoudis, E., Helander, M.E.: A single facility location problem on a tree with unreliable edges. Networks 27(3), 219–237 (1996)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions. Math. Program. 14(1), 265–294 (1978)
Shmoys, D.B., Williamson, D.P.: The Design of Approximation Algorithms, 1st edn. Cambridge University Press, New York (2011)
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Narayanaswamy, N.S., Nasre, M., Vijayaragunathan, R. (2018). Facility Location on Planar Graphs with Unreliable Links. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_23
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DOI: https://doi.org/10.1007/978-3-319-90530-3_23
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