An Improved Pareto Local Search-Based Evolutionary Algorithm for Multi-Objective Shortest-Path Network Counter-Interdiction Problem

Abstract

Most existing studies on the Shortest-Path Network Interdiction Problem (SPIP) adopt the attacker’s perspective, often overlooking the critical role of defender-oriented strategies. To support proactive defense, this paper introduces a novel problem named the Multi-Objective Shortest-Path Counter-Interdiction Problem (MO-SPCIP). The problem incorporates a backup-based defense strategy from the defender’s viewpoint and addresses the inherent trade-offs among minimizing the shortest path length, minimizing backup resource consumption, and maximizing the attacker’s resource usage. To solve this complex problem, we propose an Improved Pareto Local Search-based Evolutionary Algorithm (IPLSEA). The algorithm integrates several problem-specific components, including a tailored initial solution generation method, a customized solution representation, and specialized genetic operators. In addition, an improved Pareto Local Search (IPLS) is incorporated into the algorithm framework, allowing an adaptive and selective search. To further enhance local refinement, three problem-specific neighborhood search operations are designed and embedded within the Pareto Local Search. The experimental results demonstrate that IPLSEA significantly outperforms state-of-the-art algorithms in terms of its convergence quality and solution diversity, enabling a more robust performance in network counter-interdiction scenarios.

1. Introduction

Research on the Shortest-Path Network Interdiction Problem (SPIP) has achieved significant progress over time, particularly in modeling and algorithmic approaches from the attacker’s perspective [1]. Traditional SPIP assumes that an attacker seeks to maximize the shortest path length between designated nodes by disrupting critical network links or nodes under limited resources [2]. This attacker-centric viewpoint has found extensive applications in domains such as electrical grid analysis [3,4] and military affairs [5,6]. Notably, it serves as a valuable tool for assessing critical infrastructure vulnerability and evaluating a system’s resilience against potential disruptions.

However, a major limitation of existing SPIP studies lies in treating the network as a passive entity, overlooking the active defense strategies that are available to network defenders in real-world scenarios. In practice, network administrators possess both resources and technical capabilities to mitigate such attacks through methods like protecting key parts [7], backing up components [8,9,10], and repairing components [11]. This highlights the need for research focused on defender-oriented strategies, which offer both theoretical innovation and tangible operational benefits for real-world system protection.

This paper considers a backup-based defense strategy to mitigate the impacts of network interdiction, wherein backup links are activated to restore the shortest path connectivity following an attack. From the defender’s perspective, this involves navigating multiple competing objectives, including ensuring service availability via acceptable path lengths, minimizing the cost of deploying backup links, and maximizing the attacker’s resource expenditure by redirecting attacks toward less critical components. These objectives often conflict. For example, improving the path efficiency may demand greater resource investment, while cost reduction could compromise the performance. Moreover, effective attacker diversion requires strategic redundancy planning. These inherent trade-offs motivate our extension of the traditional SPIP into a multi-objective problem, i.e., Multi-Objective Shortest-Path Counter-Interdiction Problem (MO-SPCIP).

To solve multi-objective optimization problems like MO-SPCIP, two main kinds of approaches exist. The first kind of method is the weighted aggregation-based algorithms [12], which transform the problem into a single-objective formulation via weighted aggregation. However, the weighted aggregation-based algorithms require predefined preferences, often unavailable or uncertain in real-world settings. Another kind of method is the multi-objective evolutionary algorithms (MOEAs), which leverage Pareto dominance to generate a set of non-dominated solutions in a single run, making it more suitable for scenarios with unclear or evolving objective priorities [13]. In this paper, we adopt MOEAs due to their ability to provide a set of solutions to network administrators for flexible decisions without predefined preferences, which is more suitable for real-world applications.

Nevertheless, applying MOEAs to MO-SPCIP presents specific challenges. Particularly, MOEAs, as a class of metaheuristic algorithms, require problem-specific design adaptations to perform effectively [14]. Without such customizations, existing algorithms often struggle to generate high-quality solutions for real-world problems such as the proposed problem. On the other hand, it is still an open challenge for striking a balance between diversity and convergence during the multi-objective optimization process [15]. Ensuring the sufficient exploration of the solution space while maintaining convergence toward the Pareto front requires a carefully designed search mechanism. Therefore, we propose an enhanced MOEA named the Improved Pareto Local Search-based Evolutionary Algorithm (IPLSEA), which incorporates several problem-specific features, including tailored initial solution generation, customized solution representation, and specialized genetic operators. In addition, the algorithm integrates an Improved Pareto Local Search (IPLS) component into the framework to enhance local exploitation while maintaining global diversity.

The main contributions of this paper are as follows:

(i)

We propose a novel multi-objective optimization problem named MO-SPCIP. Different from most previous studies that focus on the attacker’s perspective, the proposed problem addresses the practical needs of network defense. By shifting the analytical focus to the defender’s perspective, the proposed problem simultaneously aims to minimize the post-attack shortest path length, reduce the consumption of defense resources, and maximize the attacker’s resource expenditure. Further, we present the mathematical model of MO-SPCIP.

(ii)

We develop an enhanced MOEA named the IPLSEA. Problem-specific components, including a tailored initial solution generation method, a customized solution representation, and specialized genetic operators, as well as an IPLS, are reasonably incorporated into the multi-objective evolutionary algorithm framework. In particular, different from the traditional Pareto Local Search (PLS) that exhaustively explores all individuals on the Pareto front, we propose an adaptive mechanism and three neighborhood search operations to make the Pareto Local Search more selective and effective.

(iii)

We conduct comprehensive experiments to evaluate the performance of the IPLSEA. The results demonstrate that the IPLSEA outperforms state-of-the-art algorithms in terms of its convergence quality, solution diversity, and approximation to the true Pareto front. Furthermore, the algorithm enables more effective backup planning strategies and significantly enhances network resilience in the face of attacks.

The rest of this paper is organized as follows: Section 2 reviews related work, Section 3 presents the model of MO-SPCIP, Section 4 details the IPLSEA, Section 5 reports the computational results, and Section 6 concludes the paper.

2. Related Works

The network interdiction problem focuses on disrupting critical network components to degrade the system performance or functionality. This research area has garnered considerable attention due to its relevance in diverse domains such as military operations [16], infectious disease containment [17], border security [18], hazardous materials transportation [19], and critical infrastructure protection in supply chains [20], power grids [4], and aviation systems [21]. Existing research primarily focuses on fundamental objectives such as shortest-path interdiction [22], maximum flow reduction [23], and connectivity degradation [24], resulting in a wide range of optimization models and algorithmic solutions.

Particularly, in the SPIP, an attacker aims to increase the shortest path length between designated source and target nodes by strategically removing or degrading links or nodes. Recent research has significantly advanced both the modeling and solution techniques for this problem. Key modeling extensions incorporate uncertainty [25], incomplete information [26], and temporal dynamics [27]. For instance, Holzmann et al. [25] developed a random attack strategy where defenders only possess probabilistic knowledge of potential attacks. Yang et al. [26] proposed a randomized feedback-based greedy strategy for sequential shortest-path interdiction under conditions of limited feedback and incomplete cost information. Bochkarev et al. [27] integrated temporal dynamics into their interdiction model. Ho et al. [28] applied a Monte Carlo Tree Search (MCTS) to optimize the balance between exploration and exploitation in strategy evaluation. Huang et al. [29] introduced a reinforcement learning framework incorporating pointer networks to manage variable-sized outputs. Ma et al. [30] developed an innovative algorithm that reformulates the SPIP as a generalized iterative set cover problem and employed an adaptive mechanism to enhance both the computational speed and solution accuracy.

While attackers dominate the focus of much of the literature, defenders of networks are rarely concerned. In the limited studies that consider the defender’s perspective, the defenders usually have access to resources and technical capabilities that enable the implementation of protective measures such as protecting key parts [7], backup components [8,9,10], and repair components [11] to mitigate the impact of attacks. For instance, Xu [7] et al. identified critical nodes to strengthen the protection of key areas, and the study demonstrates that safeguarding these nodes significantly enhances the model robustness. Banner and Orda [10] proposed the use of dedicated backup networks to protect against single-link failures, allowing for quick recovery without excessive resource allocation. Johnston et al. [8] extended this approach to handle multiple random failures. These efforts underscore the need to examine defense strategies from the defender’s perspective, especially those that incorporate pre-deployed link backups for rapid path restoration after an attack, as is the approach that this paper adopts. He and Oki [9] proposed a novel resource allocation model which provides probabilistic protection against multiple physical device failures while minimizing the required backup capacity. Moshiri et al. [11] combined hyperbolic geometry with link prediction techniques and developed a recovery strategy for complex networks to protect the network functionality and overall robustness.

Furthermore, in designing defense strategies, defenders often face competing objectives. For example, Royset [31] considered minimizing both the total interdiction cost and maximum flow in the maximum-flow network-interdiction problem, addressing this multi-objective optimization problem by decomposing it into a series of single-objective subproblems. Ramírez-Márquez et al. [32] introduced the network recovery time as a third objective and applied a probability-driven evolutionary algorithm to solve the problem. These conflicting objectives present significant challenges for the solution methods. In particular, when the number of objectives increases to three or more, the performance of traditional MOEAs often degrades due to difficulties in maintaining convergence and diversity. Many studies have been carried out to solve this issue [33,34,35,36,37]. For instance, Agrawal et al. [37] proposed a two-stage evolutionary algorithm that combines nearest neighbor and 2-opt heuristics with NSGA-II through a hybrid local search approach (HLS-EA), first generating Pareto frontier corner solutions via single-objective optimization and then using them as seeds in SHLS-EA to effectively solve multi-objective Euclidean TSP problems. Li et al. [36] proposed a Pareto front (PF) model-based local search method that first constructed a predicted PF model to identify sparse regions for targeted exploration, then integrated surrogate model optima to accelerate extreme point discovery, ultimately enhancing a surrogate-assisted multi-objective evolutionary algorithm’s convergence toward the true PF. Luo et al. [35] proposed an innovative hybrid multi-objective genetic algorithm that integrated a Pareto Local Search with greedy-based heuristic initialization and adaptive strategies to optimize truck–drone collaborative routing while balancing cost and customer satisfaction. Lv and Shen [34] developed an improved genetic algorithm with specialized local search and problem-specific operators to better optimize production and inventory scheduling for the maintenance of spare parts across three key objectives. Kolaee et al. [33] developed a novel genetic algorithm with a local search that optimized both the cost and attractiveness for medical tourism trips while considering patient preferences and time constraints.

In summary, most existing research has predominantly focused on the attacker’s perspective, with limited attention given to the defender’s viewpoint. Only a few recent studies have considered multiple optimization objectives despite their importance for addressing real-world defense scenarios. Moreover, current multi-objective optimization algorithms often struggle to balance the solution diversity and convergence, which hampers their effectiveness. To address these challenges, this paper introduces a novel multi-objective shortest-path counter-interdiction problem that simultaneously aims to minimize the post-attack shortest path length, reduce defense resource consumption, and maximize the attacker’s resource expenditure. To solve this problem, we develop a hybrid evolutionary algorithm that integrates problem-specific components with an improved Pareto Local Search. The hybrid strategy could help balance the diversity and convergence of solutions during the optimization process.

3. Problem Definition and Formulation

In this paper, we consider a backup-based defense strategy to mitigate the impacts of network interdictions. Particularly, the defender will preconfigure backup links that can be activated once an attack has occurred during the network deployment phase. After an attack, the primary network is integrated with the backup network, allowing the defender to recompute the shortest path between a source node and a target node. These backup links are treated as protected resources, meaning they are immune to attack. This also reflects real-world protection mechanisms, such as physical isolation and redundant path hardening, which are commonly used to ensure the resilience and reliability of critical communication infrastructure.

By leveraging backup links, the defender can substantially reduce the shortest path length after an attack. This concept is illustrated with a simple example below. Consider the network shown in Figure 1, where the defender attempts to transmit a message from node $s$ to node $t$. Each link is labeled with its transmission delay, and the defender’s objective is to minimize the message delivery time. Parenthetical values indicate the increased delays for traversing links when the links are attacked. The attacker’s goal is to maximize the shortest message delivery time. Before any attack, the shortest transmission path of the network is $s$ → 3 → $t$ (shown as a green solid line), with a total delay of nine. Then, as shown in Figure 2, when the attacker is allowed to interdict up to three links, the optimal attack strategy (marked by red solid lines) intercepts links ($s$, 1), ($s$, 3), and ($s$, $t$). In this context, the defender’s shortest available path becomes $s$ → 1 → $t$, resulting in a total delay of 14. Suppose the defender has preconfigured a protected backup link ($s$, 2) with a transmission delay of four, as shown in Figure 3. Since this link cannot be attacked, the attacker’s best strategy shifts to targeting (2, $t$). Consequently, the defender’s new shortest path becomes $s$ → 2 → $t$, with a total delay of 7, demonstrating how backup links can halve the post-attack delay from 14 to 7.

However, the practical deployment of backup links involves critical trade-offs, including ensuring acceptable path lengths after an attack to maintain service continuity, managing the cost of deploying and maintaining backup resources, and maximizing the attacker’s resource expenditure to divert attacks from vital components. These considerations motivate the formulation of the Multi-Objective Shortest-Path Counter-Interdiction Problem (MO-SPCIP), which aims to minimize the post-attack shortest path length, minimize the defender’s backup resource usage, and maximize the attacker’s resource consumption, simultaneously. The notations used for the formulation are detailed in

Table 1. Notation description.

Notation	Description
$G$	Initial network without backup and attack
$N$	Set of nodes in $G$
$L$	Set of links in $G$
$A$	Set of all links constituted by the nodes in $N$
$B$	Set of backup links, $B\subset A$
$r_k$	Cost required to interdict link $k$, $k\in L$
$c_k$	Delay of passing through link $k$, $k\in A$
$d_k$	Increased delay of passing through link $k$ after it is interdicted, $k\in L$
$q_k$	Resources required to backup link $k$, $k\in B$
$FS\left(i\right)$	Forward set of node $i$, $i\in N$
$RS\left(i\right)$	Backward set of node $i$, $i\in N$
$R$	Total affordable cost of interdiction
$M$	Total available amount of backup resources
$s$	Source node of the shortest path
$t$	Target node of the shortest path
$x_k$	$x_k=1$ if link $k$ is interdicted by the attacker, 0 otherwise$, k\in L$
$y_k$	$y_k=1$ if link $k$ is chosen by the defender, 0 otherwise$, k\in A$
$z_k$	$z_k=1$ if link $k$ is backed up, 0 otherwise$, k\in B$

.

In addition, to facilitate the modeling, some assumptions are provided as follows:

(i)

All problem data are fully known to both the attacker and the defender.

(ii)

The attacker acts first, selecting which links to interdict. Afterward, the defender applies the backup strategy.

(iii)

The interaction is modeled as a zero-sum game between attacker and defender.

(iv)

The game is played in a single round with no repetitions or learning.

Assumption (i) means that the network information is fully known to the attacker and the defender. This assumption is valid in domains such as smart grids and critical infrastructure, where attackers and defenders can theoretically access complete network information through monitoring or topological analysis [38]. Assumption (ii) reflects scenarios where interdiction events occur before defensive rerouting is possible, such as the coordinated sabotage of transportation or communication links, or targeted cyberattacks that disrupt network edges before rerouting protocols activate. This ordering is consistent with Stackelberg-type leader–follower dynamics commonly used in interdiction planning models [39]. Assumption (iii) simplifies the strategy analysis and makes it easier to derive Nash or optimal response solutions. In certain sudden attacks, such as a node injection causing the entire CPS chain to fail, attacks and defenses often end in a single interaction without the need for multiple rounds of adjustments [40]. Assumption (iv) allows for a focus on immediate strategic trade-offs without the complexity of iterative learning. Although the above assumptions enhance the traceability of the problem model, we have to admit that these assumptions may inevitably reduce the fidelity to certain operational contexts. For instance, network interdiction and defense often occur over multiple stages, with both sides adapting to observed actions. Assumption (iv) may limit the model’s applicability in environments with ongoing attacks or evolving defensive strategies. More realistic assumptions could be considered in further extensions of the proposed model.

The model of MO-SPCIP is shown below:

$$\mathrm{M}\mathrm{I}\mathrm{N} f_1=\underset{x_k}{\max } \underset{y_k,z_k}{\min }\sum _{k\in A}\left(c_k+x_k{d}_k\right)y_k$$

(1)

$$\mathrm{MIN} f_2={\sum }_{k\in B}{q}_k{z}_k$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{X} f_3=\sum _{k\in L}{r}_k{x}_k$$

(3)

$$\mathrm{s}.\mathrm{t}. {\sum }_{k\in FS\left(i\right)}{y}_k-{\sum }_{k\in RS\left(i\right)}{y}_k=\left\{\begin{array}{c}1 for i=s\\ 0 \forall i\in N\left\{s,t\right\}\\ -1 for i=t\end{array}\right.$$

(4)

$$y_k\le z_k,\forall k\in A$$

(5)

$$\sum _{k\in B}{q}_k{z}_k\le M$$

(6)

$$\sum _{k\in L}{r}_k{x}_k\le R$$

(7)

$$x_k\in \left\{0,1\right\},\forall k\in L$$

(8)

$$x_k=0,\forall k\in B$$

(9)

$$y_k\in \left\{0,1\right\},\forall k\in A$$

(10)

$$z_k=1, \forall k\in L$$

(11)

$$z_k\in \left\{0,1\right\}, \forall k\in \mathrm{B}$$

(12)

Objective (1) aims to minimize the shortest path between the designated source and target nodes after the network has been attacked. Objective (2) represents minimizing the backup resource consumption by the defender, while Objective (3) focuses on maximizing the resource consumption by the attacker. Constraint (4) denotes the complete path constraint. Constraint (5) enforces that the defender cannot traverse links that are not backed up. Constraint (6) imposes a limit on the total amount of backup resources that can be deployed. Constraint (7) represents the attacker’s interdiction resource consumption limitation, restricting the number of links that can be interdicted. Constraints (8) and (9) ensure that the attacker can only interdict links in $L$, because the backup links deployed by the defender cannot be interdicted. This corresponds to the protection mechanism for backups in reality. Constraint (10) defines the range of variables $y_k$. By setting $z_k$ in the original network to one through Constraints (11), we correspond with Constraint (5) so that all links in the original network can be selected for the passage. Constraint (12) indicates that defenders can choose to backup all links except those in the initial network, which corresponds to the backup strategy in reality.

4. Solution Approach

To address the proposed MO-SPCIP, this paper proposes a Pareto Local Search-Enhanced Evolutionary Algorithm (IPLSEA) to enhance both the convergence precision and distributional diversity of solutions. The framework and important components of IPLSEA are detailed in the following.

4.1. Framework of IPLSEA

Algorithm 1 outlines the procedure of the IPLSEA. The inputs of the algorithm include the target network $G$, population size $N$, maximum iteration count $T$, crossover rate $c\_r$, and mutation rate $m\_r$. The output is an approximate Pareto optimal solution set ${PF}^{*}$.

At the initialization stage, the algorithm will generate a set of the initial population based on the initial method detailed in Section 4.2 (line 1). Following the initialization, the evolutionary process iterates until reaching the maximum iteration count $T$ (lines 3–19). In each iteration, the offspring population $Q_t$ is generated using problem-specific genetic operators detailed in Section 4.3 (line 4). Then, by combining the parent population $P_t$ and offspring population $Q_t$ with the candidate population $R_t$, the algorithm selects a new population $P_{t+1}$ from the candidate population $R_t$ via nondominated sorting and crowding distance computation (lines 8–18). The nondominated sorting and crowding distance computation are commonly used techniques in multi-objective optimization algorithms such as NSGA-II, which could refer to [41], and the specific process of nondominated sorting can be seen in Section 4.3. When a set of solutions is added to the first Pareto front, an Improved Pareto Local Search (IPLS) is triggered with a probability $prob$ to refine ${PF}^{*}$, further enhancing the local convergence (line 12). The IPLS is an extension of the conventional Pareto Local Search (PLS) proposed in this study, which will be detailed in Section 4.5. The probability $prob$ increases linearly by $1/T$ per generation. This adaptive mechanism creates a dynamic exploration–exploitation balance where early phases emphasize a global search through a lower trigger probability while later phases prioritize local refinement with a higher trigger probability of an IPLS.

Algorithm 1: Framework of IPLSEA

Input: Network $G$, population size $N$, generations $T$, crossover rate $c\_r$, mutation rate $m\_r$. Output: Pareto front solutions ${PF}^{*}$. 1: Initialize$: P_1\leftarrow$Initialize Population $\left(G,N\right),{ PF}^{*}\leftarrow$Nondominated sorting $\left(P_1\right)\left[0\right],$                    $prob$ $\leftarrow$ 0 2: $t$ $\leftarrow$ 1 3: $\mathrm{while} t$ $\le T$ do 4:  $Q_t$ $\leftarrow$ Genetic operation ${(P}_t)$ 5:  $R_t$ ← $Q_t$ ∪ $P_t$ 6:  Nondominated sorting $\left(R_t\right)$ 7:  $P_{t+1}$ ← $\mathrm{\varnothing },$ $i$ ← 1 8:  while $\left|P_{t+1}\right|$ + $\left|F_i\right|$ $\le$ $N$ do 9:    if $i$ = 1 then 10:      ${PF}^{*}$ ← Update $\left({PF}^{*},F_1\right)$ 11:      if $rand$ $<$ $prob$ then 12:        ${PF}^{*}$ ← IPLS $\left({PF}^{*}\right)$ 13:      $F_1$ ← ${PF}^{\mathrm{*}}$ 14:    Crowding distance $\left(F_i\right)$ 15:    $P_{t+1}$ ← $P_{t+1}$ ∪ $F_i$ 16:    $i$ ← $i+1$ 17:  Sort $\left(F_i,\prec N\right)$ 18:  $P_{t+1}$ ← $P_{t+1}$ ∪ $F_i\left[1:N-\left|P_{t+1}\right|\right]$ 19:  $t$ ← $t+1$, $prob$ $\leftarrow$ $prob$ + 1/$T$

4.2. Initial Solution Generation

The strategy for generating initial solutions plays a critical role in obtaining high-quality outcomes. To balance the population diversity with algorithmic efficiency, we design a problem-specific initial solution generation method. In this approach, half of the initial population is generated randomly, while the other half is constructed based on key path information.

Particularly, for the random generation process, a subset of candidate backup links is randomly selected and designated as backup links. This promotes diversity in the initial population. We define a parameter $\rho ϵ\left[\mathrm{0,1}\right]$ to control the proportion of candidate backup links to be selected as backup links. For the path-based generation, we define candidate backup links that are connected to either the source node or the target node as key links. These links are likely to have a higher impact on path recovery. During the construction of each individual in this part of the population, a random number of key links is selected. This process continues until half of the initial population is generated using this strategy. Figure 4 illustrates the concept of key links, in which solid black lines represent the original network, dashed lines represent candidate backup links, and red dashed lines indicate key links used during the initial solution generation process. Furthermore, during initial solution generation, the total resource consumption of the selected backup links must not exceed the available resource constraint, ensuring that each solution is feasible with respect to the resource limits.

To facilitate computation, the solution is represented using a binary encoding scheme, where the length of the binary vector corresponds to the number of candidate backup links. Each bit in the vector indicates whether a backup link is selected (marked by 1) or not selected (marked by 0). In the example shown in Figure 5, the solution includes 10 candidate backup links, with selected links marked by 1 and non-selected links by 0.

4.3. Nondominated Sorting

Algorithm 2 introduces the procedure of nondominated sorting. The input of the algorithm is a set of individuals $P$ where each individual has $m$ objectives. The output is a set of Pareto fronts $F$ sorted by their nondomination level. First, two data structures are initialized for each individual in the population. One is the dominance set $S_i$, which keeps track of all solutions dominated by individual $p_i$. The other is the $domCount\left[i\right]$, which records how many individuals dominate $p_i$. Both structures start as empty or zero (lines 1–3). Then, the algorithm compares every pair of individuals to determine dominance based on the multi-objective rule. Specifically, if an individual $p_i$ is not worse than individual $p_j$ in all objectives and strictly better in at least one, $p_i$ is said to dominate $p_j$. In this case, $p_j$ is added to $S_i$ and $domCount\left[i\right]$ is increased by one. If $p_j$ dominates $p_i$ instead, the dominance set and domination count are updated accordingly (lines 4–10). After pairwise comparisons, the algorithm identifies the first Pareto front by selecting solutions with zero $domCount$ (lines 11–14). Next, the algorithm builds subsequent fronts iteratively. For each individual in the current front, the $domCount$ of every solution in its dominance set is decreased by one. Whenever an individual’s $domCount$ reaches zero, it is assigned to the next Pareto front. This process repeats until every individual is assigned to a front (lines 15–24).

Algorithm 2: Nondominated sorting

Input: Population $P=\left\{p_1,p_2,\dots p_n\right\}$ $\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} m \mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}$. Output: Set of Pareto fronts $F=\left[F_1,F_2,\dots \right]$ sorted by nondomination level. 1: for each $p_i$ in $P$ do 2:  $S_i\leftarrow \mathbf{\varnothing }$ 3:  $domCount\left[i\right]$ $\leftarrow$ 0 4: for each pair $\left(p_i,p_j\right)$ with $i\ne j$ do 5:  if $p_i\prec p_j$ then 6:    $S_i\leftarrow S_i\cup \left\{p_j\right\}$ 7:    $domCount\left[j\right] \leftarrow domCount\left[j\right]+1$ 8:  else if $p_j\prec p_i$ then 9:    $S_j\leftarrow S_j\cup \left\{p_i\right\}$ 10:    $domCount\left[i\right] \leftarrow domCount\left[i\right]+1$ 11:$F_1$ ← $\varnothing$ 12: for each $p_i$ in $P$ do 13:  if $domCount\left[i\right]$ = 0 then 14:    $\mathrm{a}\mathrm{d}\mathrm{d}\left(p_i\right) to F_1$ 15:$t$ ← 1 16: while $F_t$ $\ne \varnothing$ do 17:  $Q$ ← $\varnothing$ 18:  for each $p_i\in F_t$ do 19:    for each $p_j\in S_i$ do 20:      $domCount\left[j\right] \leftarrow domCount\left[j\right]-1$ 21:      $\mathrm{if} domCount\left[j\right]$ = 0 then 22:        $\mathrm{a}\mathrm{d}\mathrm{d}\left(p_j\right) to Q$ 23:  $t$ ← $t+1$ 24:  $F_t\leftarrow Q$ 25: return $F$

4.4. Genetic Operations

Through crossover and mutation operators, the algorithm could perform genetic recombination and local perturbation on individuals in the current population. Based on the solution structure introduced above, this section describes tailored crossover and mutation operations for the proposed algorithm.

(i)

Crossover

The algorithm will randomly select two parent individuals from the current population and perform the crossover operation based on these two parents. A preset crossover probability ($c\_r$) is used to decide whether to perform the crossover operation. If the execution condition is satisfied, a uniform crossover is performed on the parent individuals. Otherwise, the two parent individuals will be directly copied as offspring individuals. An example of the crossover operation is in Figure 6. For each bit of the individual’s genes (i.e., the selection state of each link), the values of the corresponding bits are exchanged between Parent 1 and Parent 2 with a probability of 0.5 to construct two child individuals.

(ii)

Mutation

To further increase the diversity of the population, we propose three kinds of mutation operations, including add_backup, remove_backup, and random_flip. The algorithm will randomly select one of the mutation operations each time during offspring generation. The illustrations of the three mutation operations are shown in Figure 7.

• add_backup

This operation randomly selects a link that is not currently backed up and attempts to add a backup to the current solution, while ensuring that the total resource constraint is not violated. Specifically, the operation traverses the set of non-backed-up links and filters out those candidate links that would still satisfy the resource constraints if included. A link from the filtered set is then randomly selected and added as a backup. This mutation operation could enhance the solution diversity and introduce local perturbations without violating the feasibility.

b.

remove_backup

This operation randomly selects a currently backed-up link and reverses its status, removing the backup designation. It is a deterministic and computationally efficient operation. By recovering potentially over-allocated resources, it helps maintain solution feasibility and creates more space for subsequent optimization. This operator is particularly useful for escaping local optima by freeing up resources for more promising backup configurations.

c.

random_flip

This operation creates a neighborhood solution by randomly selecting a link index in the solution structure and flipping its binary state (e.g., changing a 0 to 1 or a 1 to 0). Unlike the other two operations, this one does not check resource constraints, which allows it to introduce greater diversity into the population. It is especially effective during the early stages of the algorithm, when global exploration is critical for covering a diverse regions of the search space.

4.5. Improved Pareto Local Search

To maintain a balance between convergence and diversity, we propose an Improved Pareto Local Search (IPLS). Unlike traditional a Pareto Local Search [42], which exhaustively explores all individuals on the Pareto front, IPLS introduces a more selective and efficient mechanism. It is adaptively triggered during the optimization process based on a generation-dependent probability and operates exclusively on previously unexplored solutions within the current non-dominated set. Furthermore, we design three problem-specific neighborhood search operations, i.e., heuristic_add, greedy_remove, and swap_backup, to guide the local search effectively, as shown in Figure 8. For each newly examined solution, the algorithm sequentially applies these three operations to generate neighborhood solutions. The IPLS could ensure the systematic exploration of the solution space without redundant computational overhead. The three designed neighborhood search operations are introduced as follows.

• heuristic_add
  The heuristic_add operation leverages network topology information to guide the selection of backup links. It will add a key link that satisfies resource constraints to the current solution each time. To determine a reasonable key link, we define a scoring metric, as shown in Equation (13), where $Degree\left(node1\right)$ and $Degree\left(node2\right)$ are node degrees of the two nodes and $r_{edge}$ is the resource cost of a feasible and non-backed-up key link. The degree of a node indicates the number of other nodes it is connected to (i.e., its connectivity). The key link with a higher degree will be preferred. The rationale behind this metric is that high-degree nodes often serve as critical network hubs intersecting multiple shortest paths, making their connecting links highly valuable for structural resilience. By accounting for both structural significance and resource efficiency, this operation aims to maximize the structural benefit per unit of resource, thereby improving the efficiency of the backup configuration.

  $$score={\displaystyle \frac{Degree\left(node1\right)+Degree\left(node2\right)}{1+r_{edge}}}$$

  (13)
• greedy_remove
  The operation employs a greedy strategy to remove an existing backup link that contributes least to the network performance from a solution. Particularly, this operation will evaluate the impact of removing each existing backup link, where the impact is measured by the resulting increase in the shortest path length. Then, the link with minimal negative impact will be removed. This operation allows for the recovery of redundant resources while preserving the overall quality of the solution, creating space for further optimization.
• swap_backup
  The swap_backup operation performs a state exchange by randomly selecting one backed-up link and one non-backed-up link, and swapping their backup statuses. This operation does not perform feasibility verification, but it ensures that total resource consumption remains constant. By exploring different resource configurations under a fixed resource budget, this operator enables the search to uncover better-performing solutions while preserving the underlying structure of the solution space, thereby introducing diversity without violating feasibility constraints.
  The framework of the IPLS is displayed in Algorithm 3. The inputs of the algorithm include Pareto front solutions ${PF}^{*}$, maximum iterations $plsIter$, and three specific neighborhood search operators $neighOps$. The output is an improved Pareto front ${PF}^{*}$. Initially, the algorithm will identify the solutions new to ${PF}^{\mathrm{*}}$. Following the initialization, the search process iterates until reaching the maximum iteration count or no new solutions (lines 3–19). In each iteration, the algorithm performs the search operator in $neighOps$ one by one on each new solution in ${PF}_{new}^{\mathrm{*}}$ (lines 4–6). If the obtained new solution is not dominated by any solutions in ${PF}^{*}$, it will be considered to be a potential and added to $nextNew$ and ${PF}^{*}$ for the next iteration. Meanwhile, the solutions in ${PF}^{*}$dominated by the new solution will be eliminated.

Algorithm 3: IPLS

Input: Pareto front solutions ${PF}^{*}$, maximum iterations $plsIter$,         neighborhood operators $neighOps$ Output: Improved Pareto front ${PF}^{*}$. 1: Initialize: $iter$ ← 1, ${PF}_{new}^{*}$ ← Find solutions new to ${PF}^{*}$ 2: while ${PF}_{new}^{*}$ $\ne$ $\mathrm{\varnothing }$ and $iter$ $<$ $plsIter$ do 3:  $nextNew$ ← $\mathrm{\varnothing }$ 4:  for each $s$ in ${PF}_{new}^{\mathrm{*}}$ do 5:    for each $op$ $\in$ $neighOps$ do 6:      $s^{\prime }$← $op\left(s\right)$ 7:      $isDom$ ← $\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}$ 8:       for each $e$ $\in$ ${PF}^{\mathrm{*}}$ do 9:        if dominates ($e$, $s^{\prime }$) then 10:         $isDom$ ← $\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}$ 11:         break 12:       if $\neg isDom$ then 13:         for each $e$ $\in$ ${PF}^{\mathrm{*}}$ do 14:            if dominates ($s^{\prime }$, $e$) then 15:              ${PF}^{*}$ $\leftarrow {PF}^{*}\backslash \left\{e\right\}$ 16:         ${PF}^{*}$← ${PF}^{*}$ $\bigcup$ $\left\{s^{\prime }\right\}$ 17:         $nextNew$ $\leftarrow nextNew \cup \left\{s^{\prime }\right\}$ 18:  ${PF}_{new}^{*}$ $\leftarrow nextNew$ 19:  $iter$ ← $iter+$ 1 20: return ${PF}^{*}$

5. Computational Experiments

In this section, we design various typical undirected network topologies as experimental scenarios to verify the effectiveness and superiority of the proposed IPLSEA in solving MO-SPCIP through comparison experiments, ablation experiments, and parameter sensitivity analysis experiments.

5.1. Experimental Design

(i)

Instances

This study employs four characteristic undirected network topologies as experimental scenarios, as shown in

Table 2. Structural information of the networks.

Network	Number of Nodes	Number of Original Links	Number of Backupable Links
ER1	8	12	16
ER2	20	152	38
BA1	8	12	16
BA2	20	30	160
WS1	8	21	7
WS2	20	80	110
Sioux Falls	23	38	215
Eastern Massachusetts	74	129	2572

.

• Erdős–Rényi Random Network (ER)

The Erdős–Rényi (ER) network [43] is generated by connecting each pair of nodes with a uniform probability, resulting in a random graph where the degree distribution approximately follows a Poisson distribution. In this study, two ER network instances are constructed. The first instance, ER1, consists of 8 nodes with a connection probability of 0.5. The second instance, ER2, contains 20 nodes, also with a connection probability of 0.5. These random topologies provide a baseline for testing the algorithm performance in uniformly connected environments.

b.

Barabási–Albert Scale-Free Network (BA)

The Barabási–Albert (BA) network [44] is built using a preferential attachment mechanism, where new nodes are more likely to connect to existing nodes with higher degrees. This results in a scale-free topology characterized by a power-law degree distribution. Two test instances are created for this study. In BA1, the network begins with a 3-node star, and 5 nodes are added sequentially, each forming 3 connections ($m$ = 3). In BA2, the network starts with a 5-node star (node 0 as the hub), and 12 nodes are added with 2 to 4 connections per node ($m$ ≈ 3 on average). These networks model real-world systems where hubs naturally emerge.

c.

Watts–Strogatz Small-World Network (WS)

The Watts–Strogatz (WS) network [45] is constructed from a regular ring lattice, where each node is connected to its $k\_ws$ nearest neighbors, and a fraction of the links is randomly rewired with probability $p\_ws$. This results in a small-world topology that combines high clustering with short average path lengths. Two instances are used. WS1 is a 7-node torus where each node connects to 4 neighbors ($k\_ws$ = 4) with a rewiring probability of 0.3. WS2 is a 20-node torus with $k\_ws$ = 8 and $p\_ws$ = 0.3. These instances are used to evaluate the algorithm performance in networks with both local and global connectivity features.

d.

Real-World Network

The real-world networks used in this study are derived from the standard Sioux Falls transportation network dataset and a highway subnetwork extracted from the entire Eastern Massachusetts network dataset (EMS) [46]. These networks preserve the original node–link structure and realistic link weights. They serve as practical benchmarks to evaluate the algorithm’s effectiveness under real-world topological and operational constraints, offering valuable insights into its applicability beyond synthetic network models.

The initial and backup link weights of all networks are uniformly set to random integers between 1 and 12. In terms of the resource allocation, the interdicting resources and backup resources are configured in proportion according to the link weight at a ratio of 5:12 units. The incremental length after a link is interdicted is set to a random value between 1 and 5. For each instance, the total amount of backup resources is set to 30.

(ii)

Performance Metrics

Two principal metrics are employed to evaluate the algorithm performance from complementary perspectives: the inverted generational distance (IGD) [47] for convergence assessment and hypervolume (HV) [48] for diversity measurement.

• IGD

The IGD quantifies the average minimum Euclidean distance between points on the true Pareto front ($P$) and solutions in the obtained set ($Q$):

$$IGD\left(P,Q\right)={\displaystyle \frac{1}{\left|P\right|}}\sum _{v\in p}d\left(v,Q\right),$$

(14)

where $d\left(v,Q\right)$ calculates the Euclidean distance between points. Smaller IGD values indicate better convergence. For small networks (ER1, BA1, and WS1), the true Pareto front is obtained via a brute-force search. For larger networks, we construct an approximate front by combining all algorithm results and performing non-dominated sorting.

b.

HV

The HV measures the volume of the objective space dominated by the solution set with respect to a reference point $z^{*}$:

$$HV=\mathsf{\delta }\left(\bigcup _{i=1}^{\left|S\right|}{v}_i\right)$$

(15)

where $\mathsf{\delta }$ denotes the Lebesgue measure, $S$ is the non-dominated solution set, and each $v_i$ represents a solution vector. Larger HV values indicate better diversity and convergence. The reference point $z^{*}$ is determined by identifying the maximum value in each objective dimension across all normalized solutions and adding a 0.1 offset to ensure strict domination. This offset guarantees all solutions lie within the measurable hypervolume.

(iii)

Comparison Algorithms and Parameter Settings

The comparison algorithms used in this study include three classical multi-objective evolutionary algorithms, i.e., NSGA-II [41], NSGA-III [49], and SPEA2 [50], and two state-of-the-art algorithms, i.e., the Hybrid Multi-Objective Modified-NSGA-II Variable Neighborhood Search (MO-NSGA-VNS) [14] and Dual-Population-based Constrained MOEAs (DPCPRA) [51].

Based on preliminary experiments presented in Section 5.4, the parameters for each algorithm are set as shown in

Table 3. Parameter settings.

Algorithm	$\mathit{\rho }$	$\mathit{p}\mathit{l}\mathit{s}\mathit{I}\mathit{t}\mathit{e}\mathit{r}$	$\mathit{c}\_\mathit{r}$	$\mathit{m}\_\mathit{r}$
NSGA-II	/	/	0.8	0.3
NSGA-III	/	/	0.8	0.3
SPEA2	/	/	0.8	0.3
DPCPRA	/	/	0.8	0.3
MO-NSGA-VNS	/	/	0.8	0.3
IPLSEA	0.1	3	0.8	0.3

for a fair comparison, where $\rho$ is the proportion of candidate backup links selected during initial solution generation, $plsIter$ is the maximum number of iterations in the improved Pareto Local Search, $c\_r$ is the crossover rate in genetic operations, and $m\_r$ is the mutation rate in genetic operations. In addition, for all algorithms, the number of populations was 200, and the number of generations was 50. All computations were conducted on a 64-bit Windows operating system with Intel(R) Core (TM) i7-10700K, 3.8 GHz, and 64 GB RAM.

5.2. Comparison Experiments

Each algorithm was run independently 30 times in 8 experimental scenarios and the results obtained are shown in

Table 4. IGD results of six algorithms (lower is better).

Network	NSGA-II		NSGA-III		SPEA2		DPCPRA		MO-NSGA-VNS		IPLSEA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
ER1	0.0178	0.0145	0.0059	0.0061	0.0013	0.0049	0.0012	0.0039	0.0000	0.0000	0.0000	0.0000
ER2	0.0811	0.0482	0.0711	0.0700	0.1236	0.0558	0.0760	0.0646	0.0575	0.0546	0.0350	0.0414
BA2	0.0898	0.0274	0.1341	0.0257	0.081	0.0243	0.079	0.0232	0.074	0.0163	0.0249	0.002
WS1	0.0172	0.0243	0.0256	0.0582	0.0384	0.0677	0.0222	0.0581	0.0000	0.0000	0.0000	0.0000
WS2	0.1542	0.0152	0.1212	0.0226	0.1097	0.0416	0.1021	0.0384	0.0874	0.0187	0.0725	0.0075
Sioux Falls	0.1200	0.0133	0.1401	0.0037	0.1209	0.0540	0.0853	0.0527	0.0507	0.0104	0.0337	0.0160
EMS	0.1978	0.0362	0.2055	0.0069	0.1492	0.0190	0.1607	0.0331	0.1378	0.0042	0.1321	0.0187

Bolded values indicate the bestresults in each case.

and

Table 5. HV results of six algorithms (higher is better).

Network	NSGA-II		NSGA-III		SPEA2		DPCPRA		MO-NSGA-VNS		IPLSEA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
ER1	0.7385	0.0052	0.7411	0.0030	0.7445	0.0019	0.7447	0.0014	0.7450	0.0000	0.7450	0.0000
ER2	0.3136	0.0137	0.3047	0.0152	0.2936	0.0290	0.3042	0.0240	0.3070	0.0204	0.3152	0.0131
BA1	0.0756	0.0267	0.0746	0.0262	0.0795	0.0247	0.0814	0.0233	0.0931	0.0000	0.0931	0.0000
BA2	0.6820	0.0155	0.6738	0.0174	0.6928	0.0140	0.6934	0.0120	0.7000	0.0054	0.7244	0.0012
WS1	0.2553	0.0047	0.2516	0.0189	0.2480	0.0224	0.2522	0.0190	0.2587	0.0000	0.2587	0.0000
WS2	0.6803	0.0331	0.7230	0.0227	0.6911	0.0270	0.7135	0.0358	0.7484	0.0096	0.7569	0.0046
Sioux Falls	0.7643	0.0099	0.7270	0.0046	0.7669	0.0643	0.8054	0.0610	0.8429	0.0197	0.8705	0.0112
EMS	0.2423	0.0520	0.2358	0.0142	0.2669	0.0327	0.2706	0.0034	0.3154	0.0206	0.2967	0.0261

Bolded values indicate the best results in each case.

, respectively.

According to the results, the IPLSEA demonstrates consistent and significant performance advantages across most tested networks. In three small-scale networks (ER1, BA1, and WS1), the IPLSEA and MO-NSGA-VNS successfully identify an entire true Pareto front, achieving an identical and best performance in both the IGD and HV. For larger and more complex networks, IPLSEA maintains a good performance compared to benchmark and state-of-the-art algorithms. In terms of the IGD, it achieves the lowest mean values in ER2, BA2, WS2, Sioux Falls, and EMS. For instance, in BA2, it records 0.0249, significantly outperforming NSGA-II (0.0898) and SPEA2 (0.0790), demonstrating its ability to generate solution sets that better approximate the true Pareto frontier. The HV results further confirm this advantage, with the IPLSEA achieving the highest hypervolume values in all but one tested network. In addition, the IPLSEA exhibits better robustness, with minimal standard deviations across different networks. For example, in BA2, the HV standard deviation is only 0.0012 and the IGD standard deviation is 0.0020, highlighting its stable performance across multiple independent runs. These comprehensive results establish IPLSEA’s dual advantage in both the solution quality and robustness, particularly for complex network optimization problems like MO-SPCIP.

For detailed analysis, BA2 serves as an illustrative example, containing 30 main links and 180 optional backup links, with attributes detailed in

Table 6. Attributes of BA2.

No.	Node to Node	Length	Cost	d	No.	Node to Node	Length	Cost	d
1	0, 1	11	5	1	16	4, 14	15	9	4
2	0, 2	8	8	1	17	5, 15	8	10	1
3	0, 3	11	9	1	18	6, 16	10	7	4
4	0, 4	13	5	4	19	7, 16	4	11	4
5	0, 5	6	6	2	20	8, 16	7	5	1
6	1, 6	8	7	3	21	9, 17	10	5	4
7	1, 7	4	8	4	22	10, 17	11	10	3
8	1, 8	9	8	3	23	11, 17	13	5	2
9	2, 8	13	10	3	24	12, 19	13	10	3
10	2, 9	7	6	1	25	13, 18	13	7	4
11	2, 10	6	6	4	26	14, 18	4	8	4
12	3, 10	5	11	4	27	15, 18	9	9	4
13	3, 11	12	9	4	28	16, 19	15	9	1
14	3, 12	7	5	4	29	17, 19	11	10	4
15	4, 13	8	9	3	30	18, 19	14	10	1

. The cost represents resources required to interdict the corresponding link and d represents the increased delay of passing through the corresponding link. With backup resource constraints set to 30 units and pivot nodes selected as 0 (source) and 19 (destination), the network’s size precludes the exhaustive determination of true Pareto frontiers. Instead, this study constructs an approximate frontier through the non-dominated sorting of all algorithm results, yielding 41 solutions, as shown in Figure 9. The shortest path length is 9, which corresponds to backup resource 8 and interdicting resource 0, where the attacker does not interdict and, at this time, the backup path of the defender is (0, 19) and the shortest path is also (0, 19). The longest path length is 38, which corresponds to both the backup resource and interdicting resource 0 and 39, representing no backup, as shown in Figure 10, where the red solid line indicates the attacker’s interdicting link and the green solid line indicates the defender’s shortest path. The minimum value of the backup resource is 0 and the maximum value is 30. When the defense chooses to backup (7, 19), (10, 19), (3, 16), and (0, 16), all 30 backup resources are used up, as shown in Figure 11, and then the attacker interdicts the links (3, 10), (0, 2), (2, 10), and (7, 16) with 36 resources and the defender chooses (0, 16), (16, 7), and (7, 19) with the length of 17. The minimum value of interdicting resources is 0 and the maximum is 40.

The comparative results for the BA2 network are presented in Figure 12 and

Table 7. IGD results of BA2 (lower is better).

Algorithm	Mean	Std	Min	Max
NSGA-II	0.0898	0.0274	0.0527	0.1451
NSGA-III	0.1341	0.0257	0.0933	0.1896
SPEA2	0.0810	0.0232	0.0441	0.1181
DPCPRA	0.0790	0.0243	0.0441	0.1451
MO-NSGA-VNS	0.0740	0.0163	0.0573	0.1068
IPLSEA	0.0249	0.0020	0.0230	0.0276

and

Table 8. HV results of BA2 (higher is better).

Algorithm	Mean	Std	Min	Max
NSGA-II	0.6820	0.0155	0.6438	0.7056
NSGA-III	0.6738	0.0174	0.6355	0.6960
SPEA2	0.6928	0.0120	0.6744	0.7114
DPCPRA	0.6934	0.0140	0.6438	0.7114
MO-NSGA-VNS	0.7000	0.0054	0.6908	0.7053
IPLSEA	0.7244	0.0012	0.7228	0.7262

, demonstrating IPLSEA’s superior performance across both evaluation metrics. Regarding the IGD values, the IPLSEA achieves an optimal performance with a mean of 0.0249, significantly outperforming control algorithms such as the MO-NSGA-VNS (0.0740) and DPCPRA (0.070). This substantial advantage reflects IPLSEA’s enhanced approximation capability to the true Pareto frontier.

In HV metrics, the IPLSEA maintains its leading position with a mean value of 0.7244, the highest among all compared algorithms, coupled with exceptional stability evidenced by its minimal standard deviation (0.0012). This combination of a high hypervolume and low variability confirms the algorithm’s robust solution quality and reliable performance. The MO-NSGA-VNS and DPCPRA show stable but inferior results. These comprehensive results underscore IPLSEA’s dual advantage in both solution quality and algorithmic stability for multi-objective network optimization challenges.

We also provide the computational complexity analysis of the IPLSEA and its comparison algorithms. IPLSEA’s complexity is mainly determined by nondominated sorting, with a single generation being approximately $O\left({MN}^2\right)$ and the overall complexity being $O\left(T{MN}^2\right)$, where $N$ is the population size, $M$ is the number of objectives, and $T$ is the number of iterations. In comparison, the classical NSGA-II and NSGA-III also have the same dominant complexity. NSGA-III adds reference point processing, but the overall complexity remains $O\left(T{MN}^2\right)$. SPEA2 calculates the distance matrix between individuals, with a complexity of $O\left(T{MN}^2\right)$. DPCPRA adopts a two-population structure, with a complexity approximately twice that of non-dominated sorting, still belonging to the $O\left(T{MN}^2\right)$ level. MO-NSGA-VNS introduces a neighborhood search, with a complexity approximately equal to $O(T{MN}^2+NL)$, where the local search overhead $L$ is typically smaller than the non-dominated sorting overhead. Overall, the IPLSEA has comparable complexity to these algorithms, and the addition of a local search helps improve the performance without significantly increasing the computational burden.

5.3. Ablation Experiments

The ablation experiments are designed to evaluate the effectiveness of key components within the IPLSEA by comparing the performance of three algorithmic variants. These include the RIA, which removes the initial solution construction module from the IPLSEA, RIPLS, which excludes the IPLS operator from the IPLSEA, and EA, which removes both components from the IPLSEA. Each variant is compared against the full IPLSEA, as well as two strong baseline methods: SPEA2 and MO-NSGA-VNS. All algorithms are evaluated on the Sioux Falls network through 15 independent runs, using IGD and HV metrics.

Figure 13’s results demonstrate RIA’s significant superiority over the RIPLSA and EA and RIPLSA’s superiority over EA, confirming that both the structured initialization and IPLS operator enhance the algorithmic performance, with the IPLS operator showing particularly pronounced benefits. However, both variants underperform relative to the full IPLSEA, as evidenced by their degraded IGD and HV values. RIA’s notably poorer results inversely validate the critical role of the proposed IPLS operator in maintaining solution quality. These findings collectively establish that the IPLSEA’s synergistic integration of structured initialization and a targeted local search delivers an optimal performance for network optimization challenges.

5.4. Parameter Sensitivity Analysis

In order to find a suitable combination of algorithm parameters and analyze the impacts of the algorithm parameters, 10 combinations of algorithm parameters were designed based on practical experience and 30 independent experimental runs on the BA2 network were conducted. The algorithm parameters involved were as follows: $\rho$ (the proportion of candidate backup links selected during initial solution generation), $plsIter$ (the maximum number of iterations in the improved Pareto Local Search), $c\_r$ (the crossover rate in genetic operations), and $m\_r$ (the mutation rate in genetic operations). The results corresponding to different parameter combinations are shown in

Table 9. IGD results of BA2 with different combinations of algorithm parameters.

$\mathit{\rho }$	$\mathit{p}\mathit{l}\mathit{s}\mathit{I}\mathit{t}\mathit{e}\mathit{r}$	$\mathit{c}\_\mathit{r}$	$\mathit{m}\_\mathit{r}$	Average IGD Value
0.1	1	0.8	0.3	0.0566
0.1	3	0.6	0.3	0.0570
0.1	3	0.8	0.1	0.0359
0.1	3	0.8	0.3	0.0255 $\downarrow$
0.1	3	0.8	0.5	0.0341
0.1	3	0.9	0.3	0.0277
0.1	5	0.8	0.3	0.0236 $\downarrow$
0.2	3	0.8	0.3	0.0368
0.3	3	0.8	0.3	0.0398
0.05	3	0.8	0.3	0.0297

.

The results indicate that setting the value of $\rho$ to 0.1 yields a better solution set. As $plsIter$ increases from 1 to 5, the IGD values show an overall downward trend, but the improvement rate decreases significantly when $plsIter$ r exceeds 3, indicating diminishing marginal returns from additional iterations. $c\_r$ primarily affects the results at extreme values (e.g., 0.6 or 0.9), while the moderate value of 0.8 typically exhibits stable performance. $m\_r$ shows little variation within the range of 0.1–0.5, and 0.3 has a relatively balanced performance in most combinations. Overall, the parameter combination ($\rho$ = 0.1, $plsIter$ = 3, $c\_r$ = 0.8, $m\_r$ = 0.3) performs nearly as well as the optimal combination ($\rho$ = 0.1, $plsIter$ = 5, $c\_r$ = 0.8, $m\_r$ = 0.3) on the IGD, with only a minimal difference between the two. However, the former has a significant advantage in computational cost, so this combination was selected as the default configuration for experiments.

To investigate the impact of backup resource allocation on the algorithm performance, we conducted 30 independent experimental runs on the Sioux Falls network instance for each of the following backup resource levels: 5, 15, 25, 30, 35, 40, and 45 units. We identified optimal compromise solutions through a normalized distance metric relative to the ideal point [37] to analyze the effects systematically. This approach involves normalizing each objective value against the ideal point and selecting the Pareto-optimal solution with the minimal Euclidean distance as the best compromise.

Figure 14 shows the average results. Adding more backup resources can shorten the path length, but uses more backup resources. Both the path length and resource use eventually level off. The attacker’s resource use stays about the same no matter how many backup resources are available.

6. Conclusions

This paper addresses a multi-objective shortest-path interdiction problem under proactive defense strategies, where defenders pre-emptively configure backup links to mitigate attacks. The formulation simultaneously optimizes three objectives, including minimizing the post-attack shortest path length, minimizing backup resource consumption, and maximizing attacker resource utilization. To solve this challenging problem, we developed a specialized multi-objective evolutionary algorithm named the IPLSEA. The proposed algorithm features a problem-specific solution structure, initial population generation, and genetic operations. Furthermore, an IPLS based on an adaptive mechanism and multiple problem-specific neighborhood search operations is incorporated into the algorithm framework to further improve the performance of the algorithm.

A comprehensive experimental evaluation demonstrates the IPLSEA’s superiority over state-of-the-art competitive algorithms, with a dominant performance in both the IGD and HV metrics. Ablation experiments confirm that both the initial solution generation method and IPLS significantly enhance the algorithm performance, with the IPLS contributing particularly substantial improvements. Sensitivity analysis experiments reveal that increased backup resources yield progressively shorter achievable path lengths until reaching a stabilization threshold, while attacker resource expenditure remains largely independent of backup availability.

Although the IPLSEA performs well in experiments, it still faces some challenges in practical applications. First, the network topology and link state data are often incomplete or noisy, which can affect the accuracy of defense strategies. Second, as the network scale and target numbers increase, the computational load of the algorithm also rises significantly, potentially failing to meet real-time response requirements, necessitating parallel computing or hardware acceleration to enhance the efficiency. Additionally, some real-world network environments change frequently, so the algorithm must have the ability to quickly adjust and adapt. Finally, there is often information asymmetry between attackers and defenders, which increases the difficulty of the strategy formulation. Future work could focus on optimizing the computational efficiency of the algorithm and designing defense mechanisms that can adapt to dynamic and asymmetric information environments to ensure that the algorithm can be better applied to real-world scenarios.