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On Fixed-Parameter Solvability of the Minimax Path Location Problem

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Abstract

The minimax path location problem is to find a path P in a graph G such that the maximum distance \(d_G(v,P)\) from every vertex \(v\in V(G)\) to the path P is minimized. It is a well-known NP-hard problem in network optimization. This paper studies the fixed-parameter solvability, that is, for a given graph G and an integer k, to decide whether there exists a path P in G such that \(\mathop{\max}\limits_{v\in V(G)}d_G(v,P)\leqslant k\). If the answer is affirmative, then graph G is called k-path-eccentric. We show that this decision problem is NP-complete even for \(k=1\). On the other hand, we characterize the family of 1-path-eccentric graphs, including the traceable, interval, split, permutation graphs and others. Furthermore, some polynomially solvable special graphs are discussed.

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Acknowledgements

The authors would like to thank the referees for their helpful comments on improving the representation of the paper.

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  1. School of Science, Henan University of Technology, Zhengzhou, 450001, Henan, China

    Hao Lin & Cheng He

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  1. Hao Lin
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  2. Cheng He
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Correspondence to Hao Lin.

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Lin, H., He, C. On Fixed-Parameter Solvability of the Minimax Path Location Problem. Commun. Appl. Math. Comput. 5, 1644–1654 (2023). https://doi.org/10.1007/s42967-022-00238-6

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  • DOI: https://doi.org/10.1007/s42967-022-00238-6

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