| Summary: | 碩士 === 世新大學 === 資訊管理學研究所(含碩專班) === 98 === The essential p-Center problem is to determine a set of p vertices of a graph G for building facilities. The objective is to minimize the maximum access distance of clients at all vertices. Let G(V, E, l, w) be a n-vertex and m-edge graph with lengths on edges and weights on vertices. Given a graph G(V, E, l, w), a practical variant, called the Weighted Connected p-Center problem (the WCpC problem), is to find a p-center of G such that the maximum weighted access distance of clients at all vertices is minimized under the additional restriction in which requires the selected p-center induce a connected subgraph of G. If w(v) = 1, for all v in V, then the problem is abbreviated as the CpC problem. We first prove that the CpC problem is NP-Hard on planar graphs and interval graphs, respectively. Second, we propose two algorithms for the WCpC problem on trees with time-complexities O(pn) and O(n log2n), respectively, by different approaches. Meanwhile, if w(v) ? C, for all v in V, where C is a set of k numbers, for some small integer k, then another algorithm with time-complexity O(kn) is proposed. Next, the extension to graphs with forbidden vertices, called the Forbidden Weighted Connected p-Center problem (the FWCpC problem) is discussed. We show that the FWCpC problem can be also solved in O(n log2n) time. Finally, we propose an O(n) time algorithm for the FCpC problem on interval graphs with unit vertex-weights and unit edge-lengths.
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