On the non-submodularity of the problem of adding links to minimize the effective graph resistance

TL;DR

This paper investigates the problem of adding links to a network to minimize effective graph resistance, demonstrating that the problem is not submodular and that greedy algorithms lack guaranteed solution quality, even for small graphs.

Contribution

It provides a counterexample showing the non-submodularity of the problem and analyzes the limitations of greedy algorithms in this context.

Findings

Submodularity ratio approaches zero in certain graph families.

Greedy algorithm solutions can be significantly worse than optimal.

Even small graphs can exhibit a solution ratio as low as 0.878.

Abstract

We consider the optimisation problem of adding $k$ links to a given network, such that the resulting effective graph resistance is as small as possible. The problem was recently proven to be NP-hard, such that optimal solutions obtained with brute-force methods require exponentially many computation steps and thus are infeasible for any graph of realistic size. Therefore, it is common in such cases to use a simple greedy algorithm to obtain an approximation of the optimal solution. It is known that if the considered problem is submodular, the quality of the greedy solution can be guaranteed. However, it is known that the optimisation problem we are facing, is not submodular. For such cases one can use the notion of generalized submodularity, which is captured by the submodularity ratio $\gamma$. A performance bound, which is a function of $\gamma$, also exists in case of generalized submodularity. In this paper we give an example of a family of graphs where the submodularity ratio approaches zero, implying that the solution quality of the greedy algorithm cannot be guaranteed. Furthermore, we show that the greedy algorithm does not always yield the optimal solution and demonstrate that even for a small graph with 10 nodes, the ratio between the optimal and the greedy solution can be as small as 0.878.