The total effective resistance, also called the Kirchhoff index, provides a robustness measure for a graph G. We consider two optimization problems of adding k new edges to G such that the resulting graph has minimal total effective resistance (i.e., is most robust)—one where the new edges can be anywhere in the graph and one where the new edges need to be incident to a specified focus node. The total effective resistance and effective resistances between nodes can be computed using the pseudoinverse of the graph Laplacian. The pseudoinverse may be computed explicitly via pseudoinversion, yet this takes cubic time in practice and quadratic space. We instead exploit combinatorial and algebraic connections to speed up gain computations in an established generic greedy heuristic. Moreover, we leverage existing randomized techniques to boost the performance of our approaches by introducing a sub-sampling step. Our different graph- and matrix-based approaches are indeed significantly faster than the state-of-the-art greedy algorithm, while their quality remains reasonably high and is often quite close. Our experiments show that we can now process larger graphs for which the application of the state-of-the-art greedy approach was impractical before.

The total effective resistance, also called the Kirchhoff index, provides a robustness measure for a graph G. We consider the optimization problem of adding k new edges to G such that the resulting graph has minimal total effective resistance (i. e., is most robust). The total effective resistance and effective resistances between nodes can be computed using the pseudoinverse of the graph Laplacian. The pseudoinverse may be computed explicitly via pseudoinversion; yet, this takes cubic time in practice and quadratic space. We instead exploit combinatorial and algebraic connections to speed up gain computations in established generic greedy heuristics. Moreover, we leverage existing randomized techniques to boost the performance of our approaches by introducing a sub-sampling step. Our different graph-and matrix-based approaches are indeed significantly faster than the state-of-the-art greedy algorithm, while their quality remains reasonably high and is often quite close. Our experiments show that we can now process large graphs for which the application of the state-of-the-art greedy approach was infeasible before. As far as we know, we are the first to be able to process graphs with 100K+ nodes in the order of minutes.