Abstract
In this work, we consider the following problem: given a graph, the addition of which single edge minimises the effective graph resistance of the resulting (or, augmented) graph. A graph’s effective graph resistance is inversely proportional to its robustness, which means the graph augmentation problem is relevant to, in particular, applications involving the robustness and augmentation of complex networks. On a classical computer, the best known algorithm for a graph with N vertices has time complexity \(\mathcal {O}(N^5)\). We show that it is possible to do better: Dürr and Høyer’s quantum algorithm solves the problem in time \(\mathcal {O}(N^4)\). We conclude with a simulation of the algorithm and solve ten small instances of the graph augmentation problem on the Quantum Inspire quantum computing platform.
Parts of this work are heavily based on the contents of De Ridder’s master’s thesis, in particular, Sects. 2 and 4. Readers interested in a more extensive treatment of the subject matter discussed in each of these sections are referred to the thesis available at https://www.ru.nl/publish/pages/769526/z_finn_de_ridder.pdf.
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de Ridder, F., Neumann, N., Veugen, T., Kooij, R. (2019). A Quantum Algorithm for Minimising the Effective Graph Resistance upon Edge Addition. In: Feld, S., Linnhoff-Popien, C. (eds) Quantum Technology and Optimization Problems. QTOP 2019. Lecture Notes in Computer Science(), vol 11413. Springer, Cham. https://doi.org/10.1007/978-3-030-14082-3_6
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