Abstract
The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n 3) algorithm for the same. We also propose an O(n 2) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.
Similar content being viewed by others
References
Chaudhry S.S., Moon I.D., McCormick S.T.: Conditional covering: greedy heuristics and computational results. Comput. OR 14(1), 11–18 (1987)
Goldberg J.B., Lunday B.J., Smith J.C.: Algorithms for solving the conditional covering problem on paths. Naval Res. Logist. 52(4), 293–301 (2005)
Golumbic M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Horne J.A., Smith J.C.: A dynamic programming algorithm for the conditional covering problem on tree graphs. Networks 46(4), 186–197 (2005)
Horne J.A., Smith J.C.: Dynamic programming algorithms for the conditional covering problem on path and extended star graphs. Networks 46(4), 177–185 (2005)
Kratsch D.: Domination and total domination on asteroidal triple-free graphs. Discrete Appl. Math. 99(1–3), 111–123 (2000)
Moon I.D., Chaudhry S.S.: An analysis of network location problems with distance constraints. Manage. Sci. 30, 290–307 (1984)
Moon I.D., Papayanopoulos L.: Facility location on a tree with maximum distance constraints. Comput. OR 22(9), 905–914 (1995)
Pfaff J., Laskar R., Hedetniemi S.T.: Linear algorithms for independent domination and total domination in series-parallel graphs. Manage. Sci. 30, 290–307 (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sivan, B., Harini, S. & Pandu Rangan, C. On Conditional Covering Problem. Math.Comput.Sci. 3, 97–107 (2010). https://doi.org/10.1007/s11786-009-0001-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-009-0001-1