Graph Augmentation: a Quantum Algorithm for Maximising Robustness

The robustness of a network is the extent to which the network is able to continue performing well when it is subject to failures or attacks [18, p. 1]. In order to quantify a network’s robustness, all kinds of graph measures have been invented, each of which captures a different perception of robustness. This work revolves around two of them: the graph’s algebraic connectivity and its effective graph resistance. The larger the graph’s algebraic connectivity, the more robust the graph, while the larger the graph’s effective graph resistance, the less robust the graph. We asked ourselves two questions, both of which are easy to understand: given a connected graph, the addition of which single edge will maximally increase the graph’s algebraic connectivity? And, given a connected graph, the addition of which single edge will maximally decrease its effective graph resistance? Up to now, no one has detected any structure in either problem. An exhaustive search takes time O(N5), irrespective of whether robustness is defined as the graph’s algebraic connectivity or minus its effective graph resistance. Kim [30] has formulated a clever kind of search that will output the answer to the first question in time O(N3), but otherwise no algorithms have been proposed. In this work, we first attempt to discover a structure in the second problem. It has been shown that the effective graph resistance of a graph relates to the eigenvalues of the graph’s Laplacian, to the random walk on the graph, and to the number of spanning trees. We make use of the last two to slightly simplify the problem and as a result dispose of a fraction of its complexity — a lot of complexity remains. Afterwards we present the main contribution of this work: the application of a quantum search algorithm by Dürr and Høyer [16] to both problems. We show that the algorithm can be used to answer both questions in time O(N4), which means that it is the “fastest” known algorithm for solving the problem of minimising the effective graph resistance of the augmented graph. Finally, we also illustrate how heuristics can be used to speed up Dürr and Høyer’s algorithm, in particular, we explain how by taking advantage of a very simple heuristic, the query complexity of the algorithm can be halved.