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

TL;DR

This work addresses the problem of adding $k$ links to minimize the effective graph resistance, formalized via the normalized objective $r_G$, and shows that the problem is not generalized submodular as the submodularity ratio $ abla$ can vanish. The authors construct a graph family on $2N$ nodes for which $ abla$ tends to zero as $N\to\infty$, implying that greedy guarantees cannot be established; they also demonstrate that the greedy solution need not be optimal. Empirically, on small graphs with $N\le 10$, the ratio of optimal to greedy solutions can be as low as $0.878$, indicating potentially substantial suboptimality. These findings highlight the limitations of greedy approaches for $k$-GRIP and motivate developing alternative bounds or graph-structured strategies for robustness optimization.

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 $γ$. A performance bound, which is a function of $γ$, 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.

Figures (14)

• Figure 1: The smallest counterexample showing that the normalized effective graph resistance $r_G$ in $k$-GRIP is not submodular. The graph $G$ consists of 5 nodes with 5 links, denoted as solid black lines. The set $S$ is the empty set, while the element $v$ is the link between nodes 1 and 2, represented by a dashed green line. The set $R$ is the link between nodes 2 and 3, represented by a dotted red line.
• Figure 2: The graph $G$ consisting of $2N$ nodes; two path graphs with $N/2$ nodes are attached to a complete bipartite graph $K_{2,N-2}$ on $N$ nodes. The element $v$ is the dashed green link in the complete bipartite graph between node $i$ and $j$. The set $R$, represented by dotted red links, is the union of the link connecting node $i$ with the left-most node of the graph and the link connecting node $j$ with the right-most node of the graph. In this example, $N=6$.
• Figure 3: The upper bound for the submodularity ratio $\gamma$ according to Eq. \ref{['eq_counterexample_gamma']} and its asymptote $6/N$ for various values of $N$.
• Figure 4: The graph $G$ with the currently known smallest efficiency $\eta_{\min}=0.878$ on $N=10$ nodes and the $k=3$ added links are shown as dashed green links.
• Figure 5: The original graph $G$. The element $v$ is the dashed green link between node $i$ and $j$.

Theorems & Definitions (11)

• Definition 2
• Definition 3: nemhauser1978subdmodularity
• Definition 5: daskempe2011submodularityratio
• Theorem 6: bian2017curvature
• Definition 7: bian2017curvature
• Theorem 8
• proof
• Theorem 9
• proof
• Theorem 10: Theorem 2.1 from yang2013graphresistance