Facility Location on Planar Graphs with Unreliable Links

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.