Computation of inverse 1-center location problem on the weighted trapezoid graphs.

*(English)*Zbl 1452.05077Summary: Let \(T_{TRP}\) be the tree corresponding to the weighted trapezoid graph \(G=(V,E)\). The eccentricity \(e(v)\) of the vertex \(v\) is defined as the sum of the weights of the vertices from \(v\) to the vertex farthest from \(v \in T_{TRP}\). A vertex with minimum eccentricity in the tree \(T_{TRP}\) is called the 1-center of that tree. In an inverse 1-center location problem, the parameter of the tree \(T_{TRP}\) corresponding to the weighted trapezoid graph \(G=(V,E)\), like vertex weights, have to be modified at minimum total cost such that a pre-specified vertex \(s\in V\) becomes the 1-center of the trapezoid graph \(G\). In this paper, we present an optimal algorithm to find an inverse 1-center location on the weighted tree \(T_{TRP}\) corresponding to the weighted trapezoid graph \(G=(V,E)\), where the vertex weights can be changed within certain bounds. The time complexity of our proposed algorithm is \(O(n)\), where \(n\) is the number of vertices of the trapezoid graph \(G\).

##### MSC:

05C22 | Signed and weighted graphs |

05C85 | Graph algorithms (graph-theoretic aspects) |

05C05 | Trees |

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

##### Keywords:

tree-networks; center location; 1-center location; inverse 1-center location; inverse optimization; tree; trapezoid graphs##### References:

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