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On conditional covering problem. (English) Zbl 1205.05229
Summary: 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 J. A. Horne and J. C. Smith [Networks 46, No. 4, 177–185 (2005; Zbl 1093.90072)]. We show that their algorithm is wrong and further present a correct $O\left({n}^{3}\right)$ algorithm for the same. We also propose an $O\left({n}^{2}\right)$ 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.
##### MSC:
 05C85 Graph algorithms (graph theory) 05C69 Dominating sets, independent sets, cliques 05C70 Factorization, etc. 90C39 Dynamic programming 90B80 Discrete location and assignment
##### References:
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