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Dynamic programming algorithms for the conditional covering problem on path and extended star graphs. (English) Zbl 1093.90072
Summary: The Conditional Covering Problem (CCP) is a facility location problem on a graph, where the set of nodes represents demand points and potential facility locations. The key aspect of the CCP is that each facility covers all nodes within a given facility-specific coverage radius, except for the node at which it is located. The objective of this problem is to minimize the sum of the facility location costs required to cover all demand points. We first discuss the worst-case complexity of the CCP by examining literature related to the total domination problem, which is a special case of the CCP. Next, we examine the special case of path graphs and provide an $O\left({n}^{2}\right)$ algorithm for its solution. Finally, we leverage information obtained from this procedure to derive an optimal algorithm for “extended star” graphs (multiple paths having one node in common), without increasing the worst-case complexity of the algorithm.

MSC:
 90C39 Dynamic programming 90B80 Discrete location and assignment 05C69 Dominating sets, independent sets, cliques 05C70 Factorization, etc. 90B10 Network models, deterministic (optimization) 90C35 Programming involving graphs or networks