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Solving the Absolute 1-Center Problem in the Quickest Path Case

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Abstract

An undirected graph \(G=(V,A)\) by a set V of n nodes, a set A of m edges, and a set \(D\subseteq V\) consists of h demand nodes are given. Peeters (Eur J Oper Res 104:299–309, 1998) presented the absolute 1-center problem, which finds a point x placed on nodes or edges of the graph G with the property that the cost distance from the most expensive demand node to x is as cheap as possible. In the absolute 1-center problem, the distance between two nodes is computed through a shortest path between them. This paper expands the idea of Peeters (1998) and presents a new version of the absolute 1-center problem, which is called the absolute quickest 1-center problem. A value \(\sigma \) is given, and the problem finds a point \(x^*\) placed on nodes or edges of the graph G with the property that the transmission time of the quickest path to send \(\sigma \) units of data from the farthest demand node to \(x^*\) is the minimum value. We presented an \(O(r|D|(m+n\mathrm{{log}}\ n))\) time algorithm to solve the absolute quickest 1-center problem, where r is the number of distinct capacity values.

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Acknowledgements

We would like to express great appreciation to anonymous reviewers for their valuable comments and suggestions, which have helped to improve the quality and presentation of this paper.

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Correspondence to Mehdi Ghiyasvand.

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Communicated by Ebrahim Ghorbani.

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Ghiyasvand, M., Keshtkar, I. Solving the Absolute 1-Center Problem in the Quickest Path Case. Bull. Iran. Math. Soc. 48, 643–671 (2022). https://doi.org/10.1007/s41980-021-00536-4

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