Minimizing the effective graph resistance by adding links is NP-hard

TL;DR

This paper proves that optimizing the effective graph resistance of a network by adding a limited number of links is an NP-hard problem, highlighting its computational complexity.

Contribution

It establishes the NP-hardness of minimizing effective graph resistance through link addition, a key insight for network robustness optimization.

Findings

Proves the NP-hardness of the problem.

Highlights computational challenges in network design.

Provides theoretical foundation for future algorithms.

Abstract

The effective graph resistance, also known as the Kirchhoff index, is metric that is used to quantify the robustness of a network. We show that the optimisation problem of minimizing the effective graph resistance of a graph by adding a fixed number of links, is NP-hard.

Equations

$$R_G(G)=\sum _{1\le i<j\le N}{\omega }_{ij}.$$

$$R_G(G)=N\sum _{k=2}^N\frac{1}{{\lambda }_k}.$$

$$R_G(G)\ge \frac{N^2(N-1)}{2L}-1,$$

$${\omega }_{ij}$$

Full text

1 Introduction

Many network metrics have been utilised to quantify the robustness of a network, see for instance [1], [2], [11], [19], [20]. Freitas et al. [6] classify robustness metrics into three types: metrics based on structural properties, such as edge connectivity or diameter; metrics based on the spectrum of the adjacency matrix, such as the spectral radius or spectral gap; and metrics based on the spectrum of the Laplacian matrix, for instance the algebraic connectivity and the effective graph resistance. In this paper we consider the following optimization problem: how to augment a given graph $G$ by adding at most $k$ links, such that the robustness metric of the augmented network is optimal. As robustness metric we consider the effective graph resistance $R_{G}$, also known as the Kirchhoff index, see Ellens et al. [4]. The effective graph resistance not only covers the shortest path between any pair of nodes, but incorporates all paths between any two nodes. Because in addition $R_{G}$ decreases upon the addition of an edge to the graph [9], this makes the effective graph resistance a good metric to evaluate the robustness of a network.

Predari et al. refer to the optimization problem at hand as $k$-Graph Robustness Improvement Problem (k-GRIP) [18], in which one has to decide where $k$ links are to be added to a given network $G$, such that the robustness metric is optimised. Several researchers investigated $k$-GRIP for specific robustness metrics. For instance, Wang et al. [21] considered $1$-GRIP, with as robustness metric the second-smallest eigenvalue of the Laplacian matrix, which was coined the algebraic connectivity by Fiedler [5]. They suggest several strategies to decide which single link to add to the network, in order to increase the algebraic connectivity as much as possible. A nice overview of $k$-GRIP for the algebraic connectivity is presented in [12]. The NP-hardness of $k$-GRIP for the algebraic connectivity was proved in [14].

For the effective graph resistance, 1-GRIP was considered by Wang et al.  [22]. They investigated different strategies, based upon topological and spectral properties of the graph, to determine the most optimal link to add, and derived a lower bound for $R_{G}$ after adding a single link. Pizzuti et al. [16],  [17] proposed and evaluated several genetic algorithms to find the most optimal edge to add, in order to minimize $R_{G}$. Clemente et al. [3] studied $k$-GRIP for the effective graph resistance and gave lower bounds for $R_{G}$ upon the addition of $k$ links, under some mild conditions for $k$. For $k=1$ the lower bound in [3] clearly outperforms the lower bound in [22]. Predari et al. [18] also consider $k$-GRIP for the effective graph resistance. They focus on heuristics for $k$-GRIP based upon sampling and a fast approximation method to compute the effective graph resistance.

Although for some choices of the robustness metric, $k$-GRIP is known to be NP-hard, to the best of our knowledge this has not been proved yet for the effective graph resistance. The aim of this paper is to prove that augmenting a given graph $G$ by adding $k$ links, in order to minimize the effective graph resistance, is NP-hard.

2 Definitions and main result

In this paper we consider undirected, connected simple graphs $G=(V,E)$ without self-loops. Here $V$ denotes the set of $N$ vertices, while $E$ is the set of $L$ links connecting vertex pairs of $V$. The notation $i\sim j$ indication that nodes $i$ and $j$ are adjacent in $G$. We let $G^{c}=(V,E^{c})$ denote the complementary graph of $G$, where $E^{c}=\{(u,v)|u,v\in V,u\neq v,(u,v)\not\in E\}$. The adjacency matrix $A$ of $G$ is an $N\times N$ symmetric matrix with elements $a_{ij}$ that are either 1 or 0 depending on whether there is a link between nodes $i$ and $j$ or not. The Laplacian matrix $Q$ of $G$ is an $N\times N$ symmetric matrix $Q=\Delta-A$, where $\Delta=diag(d_{i})$ is the $N\times N$ diagonal degree matrix with the elements $d_{i}=\sum_{j=1}^{N}a_{ij}$. The eigenvalues of $Q$ are all real and non-negative and can be ordered as $0=\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{N}$.

Interpreting the graph $G$ as an electrical network whose edges are resistors of $1\Omega$, the effective resistance $\omega_{ij}$ between node $i$ and $j$ can be computed based on Kirchoff’s circuit laws. Then the effective graph resistance $R_{G}$, also known as the Kirchhoff index, is defined as the sum of the resistances over all node pairs [10]

[TABLE]

Ellens et al. [4] showed that the effective graph resistance can also be computed using the Laplacian eigenvalues $\lambda_{k}$ of the graph $G$ as

[TABLE]

Ellens et al. [4] argued that the effective graph resistance is an appropriate robustness metric. Note that the smaller the value of $R_{G}$ the larger the robustness of the network. The smallest value of the effective graph resistance for a graph on $N$ nodes is obtained for the complete graph $K_{N}$ and satisfies $R_{G}(K_{N})=N-1$. We will show in this paper that adding a specified number of edges to a given graph, in order to minimize the effective graph resistance, is NP-hard. We will now give an explicit description of the considered optimization problem.

Problem 1** (Minimum effective graph resistance augmentation problem).**

Given an undirected, connected, simple graph $G=(V,E)$, a non-negative integer $k$ and a non-negative threshold $t$, is there a subset $B\subseteq E^{c}$ of size $|B|\leq k$ such that the graph $H=(V,E\cup B)$ satisfies $R_{G}(H)\leq t$?

Problem 1 is clearly in NP, because given a graph $G$ and the set of added edges $B$, the correctness of the given solution can be verified by computing the eigenvalues of the Laplacian matrix, which is an $\mathcal{O}(N^{3})$ operation. Then simply computing (2) and comparing the outcome with the given threshold $t$ verifies the solution. Thus the minimum effective graph resistance augmentation problem is in NP.

Problem 1 is the decision version of the following optimisation problem: Given an undirected, connected, simple graph $G=(V,E)$ and a non-negative threshold $t$, find a set of currently non-existent edges of minimum size to add to $G$ such that the effective graph resistance $R_{G}$ of the augmented graph is at most $t$. We prove in this work that Problem 1 is NP-hard, which immediately implies that the corresponding optimisation problem is also NP-hard. Thus, the problem of adding a specified number of edges to a graph to minimize the effective graph resistance is also NP-hard. We now state the main result of the paper.

Theorem 2**.**

The minimum effective graph resistance augmentation problem is NP-hard.

3 Proof of Theorem 2

The proof of Theorem 2 heavily relies on the proof of the NP-hardness of the maximum algebraic connectivity augmentation problem, as given in [14]. The proof is by reduction of our augmentation problem to a problem for which NP-hardness has been proved, namely the 3-colorability problem, see [7]. For our proof we will use a construction and a lemma from [14] and two additional lemma’s.

Construction**.**

[14]** Given a graph $G=(V,E)$ with $n>1$ vertices and $m$ edges, a graph $G^{\prime}=(V^{\prime},E^{\prime})$ is constructed which consists of three disjoint copies $G_{0},G_{1}$ and $G_{2}$ of $G$. This implies that each vertex $v\in V$ is copied to a vertex $v_{i}\in G_{i}$ and each edge $(u,v)\in E$ is copied to $(u_{i},v_{i})\in G_{i}$, for $i=1,2,3$. By construction the graph $G^{\prime}$ has $3n$ vertices and $3m$ edges. We now consider the minimum effective graph resistance augmentation problem on $G^{\prime}$ with $k=3n^{2}-3m$, such that the augmented graph $H$ has at most $3n^{2}$ edges and $t=\frac{9n-5}{2}$.

Now, let $K_{n,n,n}$ denote the complete tripartite graph. In order to prove that the minimum effective graph resistance augmentation problem can be reduced to the 3-colorability problem, we will use the following three lemmas.

Lemma 3**.**

[14]** There exists a subset $B\subseteq(E^{\prime})^{c}$ of size $|B|\leq k$ such that $H=(V^{\prime},E^{\prime}\cup B)$ is (isomorphic to) $K_{n,n,n}$ if and only if $G$ is 3-colorable.

Lemma 4**.**

[13]** Let G be a simple connected graph with $N\geq 2$ vertices and $L$ edges. Then

[TABLE]

with equality if and only if $G\cong K_{N}$, or $G\cong K_{N/2,N/2}$, or $G\in\Gamma_{d}$.

Here, $\Gamma_{d}$ denotes a special class of $d$-regular graphs defined in [15]. Let $M(i)$ be the set of all neighbours of the vertex $i$, that is, $M(i)=\{k|k\in V,k\sim i\}$, where $V$ denotes the set of vertices of the graph. Then for every $1\leq d\leq n-1$ the set $\Gamma_{d}$ denotes the set of all $d$-regular graphs with diameter 2 and satisfying $|M(i)\cap M(j)|=d$ for every pair of vertices $i,j$ that are not adjacent, i.e. $i\not\sim j$.

Lemma 5**.**

A graph $H=(V,E)$ with $N=3n$ vertices and $L\leq 3n^{2}$ edges for $n>1$ satisfies $R_{G}(H)\leq\frac{9n-5}{2}$ if and only if $H$ is (isomorphic to) $K_{n,n,n}$.

Proof.

First, we compute the effective graph resistance $R_{G}$ of the complete tripartite graph $K_{n,n,n}$ using Eq. (1). Gervacio [8] derived the effective resistance between nodes in complete multipartite graphs as:

[TABLE]

where $m_{i}$ and $m_{j}$ represent the size of the partition of node $i$ and $j$ respectively. In our case, $N=3n$ and $m_{i}=m_{j}=n$. The number of node pairs in the same partition equals $3n(n-1)/2$, such that the number of pairs outside of the same partition equals $3n^{2}$. Then the effective graph resistance of the complete tripartite graph exactly equals $R_{G}(K_{n,n,n})=\frac{9n-5}{2}$.

Next, using $N=3n$ and $L\leq 3n^{2}$, according to Lemma 4 it follows $R_{G}(H)\geq\frac{9n^{2}(3n-1)}{6n^{2}}-1=\frac{9n-5}{2}$. By the condition $R_{G}(H)\leq\frac{9n-5}{2}$, we deduce that $R_{G}(H)=\frac{9n-5}{2}$. Also it follows that $L=3n^{2}$ because $N=3n$ and $L<3n^{2}$ would imply $R_{G}(H)>\frac{9n-5}{2}$ according to Lemma 4. Therefore the average degree of $H$ equals $2n$. Since $R_{G}(H)$ is equal to the lower bound given in Lemma 4, $H$ is either the complete graph $K_{3n}$, the complete bipartite graph $K_{3n/2,3n/2}$ or it is a $2n$-regular graph belonging to the class $\Gamma_{2n}$. First, assume $H\cong K_{3n}$. However, the number of links of $K_{3n}$ equals $\frac{3n(3n-1)}{2}$ which, for $n>1$, is larger than $3n^{2}$, the number of links of $H$. Therefore $H\not\cong K_{3n}$. Next assume $H\cong K_{3n/2,3n/2}$, which can only hold for $n$ even. Then the number of links of $K_{3n/2,3n/2}$ equals $\frac{9n^{2}}{4}$ which is always smaller than $3n^{2}$, the number of links of $H$. Therefore $H\not\cong K_{3n/2,3n/2}$. Hence we conclude that the graph $H$ is $2n$-regular and belongs to the class $\Gamma_{2n}$.

We will now show that $H$ is isomorphic to $K_{n,n,n}$. We start with an arbitrary node of $H$ and label it as node 1. Because $H$ is $2n$-regular, node 1 has exactly $2n$ neighbours, see Fig. 1a.

The remaining $n-1$ nodes, other than node 1 and its $2n$ neighbours, cannot be adjacent to node 1 because it already has degree $2n$, by construction. We now label these nodes as nodes 2 until $n$, see Fig. 1b. Now, because $H$ belongs to the class $\Gamma_{2n}$ and nodes 2 until $n$ are not adjacent to node 1, each of the nodes 2 until $n$ has exactly the same neighbours as node 1, see Fig. 1c.

Next, take an arbitrary node outside the set $\{1,2,\cdots,n\}$ and label it as $n+1$. To obtain degree $2n$, node $n+1$ needs to be adjacent to $n$ nodes outside the nodes $\{1,2,\cdots,n\}$. We label this set of $n$ adjacent nodes as $\{2n+1,\cdots,3n\}$, see Fig. 2a.

Finally, every node not in $\{1,2,\cdots,n+1\}\cup\{2n+1,\cdots,3n\}$ needs to share with node $n+1$ its neighbours $\{2n+1,\cdots,3n\}$, see Fig. 2b.

Denote by $S_{i}$ the nodes labelled as $\{n(i-1)+1,n(i-1)+2,\cdots,n(i-1)+n\}$, for $i=1,2,3$. Then $|S_{i}|=n$, every node pair within $S_{i}$ is not adjacent and for every $i\neq j$ all nodes in $S_{i}$ are adjacent to all nodes in $S_{j}$. This proves that $H$ is a complete tripartite graph $K_{n,n,n}$. ∎

Finally, Theorem 2 follows from combining Lemma 3 and 5.

Statements & Declarations

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Data availability

No data was used for the research described in the article.