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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-90530-3_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90529-7
Online ISBN: 978-3-319-90530-3
eBook Packages: Computer ScienceComputer Science (R0)