Abstract
We consider the problems of determining the metric dimension and the minimum cardinality of doubly resolving sets in n-cubes. Most heuristics developed for these two NP-hard problems use a function that counts the number of pairs of vertices that are not (doubly) resolved by a given subset of vertices, which requires an exponential number of distance evaluations, with respect to n. We show that it is possible to determine whether a set of vertices (doubly) resolves the n-cube by solving an integer program with O(n) variables and O(n) constraints. We then demonstrate that small resolving and doubly resolving sets can easily be determined by solving a series of such integer programs within a swapping algorithm. Results are given for hypercubes having up to a quarter of a billion vertices, and new upper bounds are reported.
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Acknowledgements
The author would like to thank Issoufou Abdou Amadou and Gilles Éric Zagré for initiating the idea of solving an integer program to determine if a set of vertices (doubly) resolves the n-cube. Special thanks go to Serge Bisaillon for his invaluable help with the use of CPLEX.
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Hertz, A. An IP-based swapping algorithm for the metric dimension and minimal doubly resolving set problems in hypercubes. Optim Lett 14, 355–367 (2020). https://doi.org/10.1007/s11590-017-1184-z
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DOI: https://doi.org/10.1007/s11590-017-1184-z