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The total effective resistance, also called the Kirchhoff index, provides arobustness measure for a graph $G$. We consider two optimization problems ofadding $k$ new edges to $G$ such that the resulting graph has minimal totaleffective resistance (i.e., is most robust) -- one where the new edges can beanywhere in the graph and one where the new edges need to be incident to aspecified focus node. The total effective resistance and effective resistancesbetween nodes can be computed using the pseudoinverse of the graph Laplacian.The pseudoinverse may be computed explicitly via pseudoinversion; yet, thistakes cubic time in practice and quadratic space. We instead exploitcombinatorial and algebraic connections to speed up gain computations in anestablished generic greedy heuristic. Moreover, we leverage existing randomizedtechniques to boost the performance of our approaches by introducing asub-sampling step. Our different graph- and matrix-based approaches are indeedsignificantly faster than the state-of-the-art greedy algorithm, while theirquality remains reasonably high and is often quite close. Our experiments showthat we can now process larger graphs for which the application of thestate-of-the-art greedy approach was impractical before.

Greedy Optimization of Resistance-based Graph Robustness with Global and Local Edge Insertions