Faster Greedy Optimization of Resistance-based Graph Robustness

Abstract

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.