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

TL;DR

This paper develops faster greedy algorithms for optimizing graph robustness by adding edges to minimize total effective resistance, enabling the handling of larger graphs with improved efficiency.

Contribution

It introduces combinatorial and algebraic techniques combined with randomized sampling to accelerate edge addition strategies for graph robustness optimization.

Findings

Faster algorithms outperform existing methods in speed.

Approaches scale to larger graphs previously infeasible.

Resulting graphs maintain high robustness quality.

Abstract

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.