Reliability Analysis of Survivable Networks under the Hostile Model ^†

Abstract

This article studies the Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability and introduces a metaheuristic resolution approach based on Greedy Randomized Adaptive Search Procedure and Variable Neighborhood Descent. Under the hostile model, nodes and links are subject to probabilistic failures. The research focuses on studying the relationship between the optimization and the reliability evaluation in a symmetric network design problem. Relevant research questions are addressed, linking the number of feasible networks for the full probabilistic model, the sensitivity with respect to elementary probabilities of operation for both edges and nodes, and the sensitivity of the model with respect to the symmetric connectivity constraints defined for terminal nodes. The main result indicates that, for the hostile model, it is better at improving the elementary probabilities of operation of links than improving the elementary probabilities of Steiner nodes, to meet a required reliability threshold for the designed network.

1. Introduction

Currently, the backbone of the infrastructure that provides support to the Internet is based on optic fiber. The internet interconnects several types of networks, providing support for large, medium, and small retail companies, Internet Service Providers, large data centers [1]; distributed datacenters [2]; and Content Delivery Networks [3], among other relevant networks. In turn, services based on optic fiber to the home have a large penetration throughout the world, and provide high data rates to the final customers. In this context, relevant issues arise regarding the network design. For example, a specific concern is that the physical design of the optic fiber network is not robust against large-scale natural disasters and/or malicious attacks [4].

In the last twenty years, both the technological and research communities have been searching for smart strategies for the augmentation of physical networks [5,6]. The importance of the fiber-optics network deployment problem led to several relevant topological network design problems. The main goal is interconnecting distinguished nodes of the network with acceptable redundancy, while simultaneously meeting reliability constraints.

Reliability analysis refers to the study of probabilistic failures on the components of a system [7]. Reliability is defined as the the capability of a system of being trustworthy or performing correctly and consistently. The reliability is a probabilistic measure of the correct operation of the system, subject to random failures [7]. In network reliability analysis, systems are modeled as graphs where the nodes are the elements of interest of the system and the edges are the relationship between said elements (e.g. electricity transmission lines, fiber optics, etc.). The operating probabilities of the elementary components of the system (nodes and/or links) are assumed to be known. After modeling the system, it is defined what an operating state is, and the reliability of the system (i.e., the probability that the system works given that failures occur in its elementary components) is computed [8].

Network reliability can be improved through diversity enhancement [9]. The main idea consists in optimizing the network design to guarantee that communications do not fail, even when node/link failures occur (within a given tolerance). The augmentation proposes including redundant paths to guarantee reliability [10].

In this line of work, this article introduces the Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability (GSPNCHR) and describes a metaheuristic resolution approach based on Greedy Randomized Adaptive Search Procedure (GRASP) and Variable Neighborhood Descent (VND). The GSPNCHR is a relevant symmetric network design problem, characterized by symmetric connectivity requirements between distinguished terminal nodes and also symmetric costs for links. The hostile model is studied, assuming that nodes and links are subject to probabilistic failures [11]. The main goal of the research is studying the relationship between the network optimization and reliability evaluation for a relevant symmetric network design problem. The study provides a useful insight to characterize the reliability of symmetric networks.

The main questions that guided the research are:

• Research Question 1 (RQ 1):How many feasible networks exists given the full probabilistic model, with input parameters $(p_{min},P_E,P_{V-T})$, where $p_{min}$ is a lower bound for the reliability of the designed network, $P_E$ is the vector of elementary probabilities of operation for links in E, and $P_{V-T}$ is the vector of elementary probabilities of operation for nodes in $V-T$ (i.e., Steiner nodes) de Steiner). The main goal is analyzing the number of solutions found by the proposed GRASP/VND metaheuristic to solve the GSPNCHR have a larger reliability under the hostil model than the established threshold $p_{min}$.
• Research Question 2 (RQ 2):What is the sensitivity of the model with respect to elementary probabilities of operation for both edges and nodes? Given the minimum reliability threshold for the network $p_{min}$, the goal is analyzing the solutions computed by the proposed GRASP/VND metaheuristic, determining if they provide a reliability larger than $p_{min}$ in two scenarios: (i) when vector $P_E$ is fixed and values of vector $P_{V-T}$ vary in a given range, and (ii) when components of the vector $P_{V-T}$ are fixed and values of vector $P_E$ vary in a given range.
• Research Question 3 (RQ 3):What is the sensitivity of the model with respect to the connectivity constraints defined for terminal nodes? Given a triplet $(p_{min},P_E,P_{V-T})$ and considering a given baseline network topology, different input networks are generated for the GSPNCHR, considering different number of terminal nodes and different connectivity requirements for each pair of terminal nodes (i.e., number of disjoint path between them). Then, the number of solutions computed by the proposed GRASP/VND metaheuristic whose reliability is over $p_{min}$ is determined.
• Research Question 4 (RQ 4):Is it better to improve the elementary probabilities of operation of links, or the the elementary probabilities of Steiner nodes, in order to meet a required reliability threshold for the designed network?

Regarding the applied methodology, on the one hand, GRASP and VND were chosen for the design and optimization due to their known efficiency in solving combinatorial optimization problems on graphs. On the other hand, the Recursive Variance Reduction (RVR) technique was used for evaluating the reliability of the designed networks, considering its better efficacy and computational efficiency with respect to other reliability estimation techniques in different scenarios [12]. RVR is characterized by returning better estimates to the exact value for the reliability of a network (in the studied problem variant, under the hostile model) and lower variance than other-well known techniques [12]

The calculation of the reliability of a network in the hostile model is an NP-hard problem. The main goal of the research is to analyze the trade-off between reliability and design cost when adding different redundancy levels between distinguished nodes. The main results indicate that the proposed approach is able to compute good quality solutions (i.e., low cost solutions) for the GSPNCHR, also complying with high reliability levels, through a specific constraint that discards feasible solutions whose reliability is lower than a certain pre-established bound ($p_{min}$).

The article extends our previous conference publication “A GRASP/VND heuristic for the Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability” [13] at 8th International Conference on Variable Neighborhood Search Abu Dhabi, United Arab Emirates, 2021. New content and contributions in this article include:

(i)

An improved definition and description of the Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability (GSPNCHR).

(ii)

An exact formulation of the GSPNCHR as a mathematical programming problem.

(iii)

A proof that the GSPNCHR is in the NP-hard complexity class.

(iv)

An extended review of related works, properly contextualized with the proposed research.

(v)

The design of a GRASP/VND hybrid metaheuristic to solve the GSPNCHR, using the RVR method for estimating the reliability of the computed networks, to be compared with the established lower bound $p_{min}$.

(vi)

An improved experimental analysis and results that highlight the fact that the proposed model is more robust under node failures than over link failures.

(vii)

Theoretical results for particular cases of the GSPNCHR are introduced.

The article is organized as follows. The next section presents a formal description for the GSPNCHR problem and demonstrates its NP-hardness. Section 3 reviews relevant related work. A GRASP/VND solution is introduced in Section 4. Numerical results are presented in Section 5. In this section the capabilities of the designed GRASP/VND and reliability evaluation approach for the GSP-NC are evaluated. Moreover, a sensitivity analysis is introduced with the aim to evaluate the impact of perturbations in the elementary reliabilities on edges and Steiner nodes. Section 6 particular cases of the addressed problem are analyzed. Among them are cases in which the problem can be solved in polynomial computational order. Finally, Section 7 presents the main conclusions of the research and formulates the main lines for future work.

2. The Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability

This section describes the GSPNCHR, its mathematical formulation as optimization problem, and the computational complexity of the problem.

2.1. Problem Model and Definition

The considered problem model proposes a network scenario where component failures may occur. In particular, the model works under the general hypothesis that both the network links and the network nodes can fail. Two types of nodes are considered, the so-called terminal nodes that model the more interesting nodes, for which connectivity requirements are set, and Steiner nodes, optional nodes that may or may not be part of feasible solutions. The connectivity requirements defined for terminal nodes are symmetric. Using Steiner nodes usually contributes to reduce the cost of the proposed network design, especially when considering connectivity requirements. Steiner nodes also have an impact on the reliability of feasible solutions.

Terminal nodes are always part of feasible solutions and different levels of connectivity between them are established in advance (i.e., a certain number of node-disjoint paths that connect them). Connectivity constraints are formulated in a a matrix of node-connectivity levels required between pairs of terminal nodes. The matrix that defines the connectivity requirements for terminal nodes is symmetric. Since both connectivity requirements and connectivity costs are symmetric, the network design problem in symmetric. The elementary probabilities of operation of the Steiner links and nodes are assumed to be known. They are independent and can have heterogeneous values, within specific ranges for both sets. Other inputs of the problem are two vectors with the probabilities of operation of the links and the probabilities of operation of the Steiner nodes. Terminal nodes are assumed to be perfect (they operate with probability 1).

Network links have associated costs that model the cost of commissioning the connection between two nodes, typically the dredging to lay a line that connects two sites. An input matrix establishes the connection costs between every pair of network nodes. The matrix that establishes the connections cost is symmetric. The goal of the problem is designing a minimum cost sub-network of the input network topology that satisfies the matrix of node-connectivity requirements between pairs of nodes and also the reliability of the designed sub-network exceeds a certain pre-established threshold, which is also an input to the problem.

The model applied to evaluate the reliability of the network is known as hostile model. In the hostile model, except for terminal nodes, any other component of the network is subjected to failures. In the proposed problem, both the Steiner nodes (non-terminal nodes) and the edges are subjected to failures with pre-established probabilities of operation. These probabilities are independent of each other and also within the set of edges and within the set of Steiner nodes. Usually the values they take are high, for example greater than 0.95.

The exact calculation of the reliability of a network in the hostile model is an NP-hard problem itself. Even when the probabilities of operation for Steiner nodes are equal to 1, the k-terminal-reliability model arises, a well-known NP-hard problem [14]. In this article, the RVR technique is adapted to the hostile model and applied to efficiently estimate the reliability of the network.

The proposed model does not consider dependencies between components. Links and Steiner nodes have probabilities of failure (or operation) independent of each other. This is a common assumption in edge-reliability, node-reliability, and also in the studied hostile model. Assuming dependencies between the failure of components requires having knowledge of the probability distributions of joint failures, which is usually not known. Models that assume failure dependencies do not necessarily adapt to different realities. We address the design of generic networks that can be instantiated in different contexts, so no assumptions about failure dependencies are considered.

Overall, the proposed model combines the concepts and goals of ‘designing a survivable network’ and ‘designing a reliable network’ in a single low-cost robust network design approach, without including non-realistic simplifications.

2.2. Mathematical Formulation

The mathematical formulation of the GSPNCHR is presented next.

Definition 1 (GSPNCHR).

Consider a simple undirected graph $G=(V,E)$, a distinguished set of terminal nodes $T\subseteq V$ (called terminals), a matrix $C={\left\{c_{i,j}\right\}}_{(i,j)\in E}$ with link-costs that models the costs of connecting pairs of network nodes, and connectivity requirements defined for each pair of terminal nodes $R={\left\{r_{i,j}\right\}}_{i,j\in T}$. Links are assumed to fail with probability $P_E={\left\{p_e\right\}}_{e\in E}$ and nodes are assumed to fail with probability $P_{V-T}={\left\{p_v\right\}}_{v\in V-T}$. The goal of the problem is building a minimum-cost topology $G_S\subseteq G$ meeting both the connectivity requirements R and a given reliability threshold $p_{min}$ i.e., $R_K\left(G_S\right)\ge p_{min}$, being $K=T$ the set of terminal nodes. $R_K\left(G_S\right)$ denotes the probability that the graph $G_S$ spans the terminal set $K=T$, where links and Steiner nodes may fail with probabilities $P_E$ and $P_{V-T}$, respectively. Namely, $R_K\left(G_S\right)$ is the reliability of $G_S$ under the hostile model.

The mathematical formulation of the GSPNCHR considers the following elements:

• A graph $(G,C,R,T,P_E,P_{V-T})$, according to Definition 1.
• A value $p_{min}$, which determines a lower bound of reliability to be satisfied by the feasible solutions of the problem.
• A decision variable $y_{(i,j)}^{u,v}$, defined in Equation (1), that indicates if the link $(i,j)\in E$, directed from i to j, is used on a path from the terminal node u to the terminal node v in the built solution.

  $$y_{(i,j)}^{u,v}=\left\{\begin{array}{cc}1\hfill & \mathit{in}\mathit{case}(i,j)\in E\mathit{is}\mathit{used}\mathit{in}\mathit{a}\mathit{path}u-i-j-v\hfill \\ 0\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

  (1)
• A decision variable $x_{(i,j)}$, defined in Equation (2), that indicates if the (undirected) link $(i,j)$ is used in the built solution.

  $$x_{(i,j)}=\left\{\begin{array}{cc}1\hfill & \mathit{in}\mathit{case}(i,j)\in E\mathit{is}\mathit{used}\mathit{in}\mathit{the}\mathit{solution}\hfill \\ 0\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

  (2)

The mathematical formulation of the GSPNCHR as a combinatorial optimization problem is presented in Equations (3)–(12).

$$\begin{array}{ccc}\hfill min& \sum _{(i,j)\in E}{c}_{i,j}{x}_{i,j}\hfill & \hfill \left(3\right)\\ \hfill \mathrm{subject}\mathrm{to}& x_{ij}\ge y_{(i,j)}^{u,v}+y_{(j,i)}^{u,v}\forall (i,j)\in E,\forall u,v\in T,u\ne v\hfill & \hfill \left(4\right)\\ \\ \hfill & \sum _{(u,i)\in E}{y}_{(u,i)}^{u,v}\ge r_{u,v}\forall u,v\in T,u\ne v\hfill & \hfill \left(5\right)\\ \\ \hfill & \sum _{(j,v)\in E}{y}_{(j,v)}^{u,v}\ge r_{u,v}\forall u,v\in T,u\ne v\hfill & \hfill \left(6\right)\\ \\ \hfill & \sum _{(i,p)\in I\left(p\right)}{y}_{(i,p)}^{u,v}-\sum _{(p,j)\in I\left(p\right)}{y}_{(p,j)}^{u,v}\ge 0,\forall p\in V-\{u,v\},\forall u,v\in T,u\ne v\hfill & \hfill \left(7\right)\\ \\ \hfill & \sum _{(s,i)\in E}{x}_{s,i}\le M{\widehat{x}}_s,\forall s\in V-T\hfill & \hfill \left(8\right)\\ \\ \hfill & R_K\left(G_S\left(\left\{x_{i,j}\right\}\right)\right)\ge p_{min}\hfill & \hfill \left(9\right)\\ \\ \hfill & x_{(i,j)}\in \{0,1\}\forall (i,j)\in E\hfill & \hfill \left(10\right)\\ \\ \hfill & {\widehat{x}}_i\in \{0,1\}\forall i\in V-T\hfill & \hfill \left(11\right)\\ \\ \hfill & y_{(i,j)}^{u,v}\in \{0,1\}\forall (i,j)\in E,\forall u,v\in T,u\ne v\hfill & \hfill \left(12\right)\\ \hfill \end{array}$$

The objective function is expressed in Equation (3). It proposes minimizing the overall design cost of the reliable network design, considering all links included in a solution and the cost function defined for edges of the graph G.

The constraints of the problem are defined in Equations (4)–(12). Constraints in Equation (4) state that the considered network links are one-way, i.e., a direct graph is considered to model the communication network. It also establishes that the undirected edge $(i,j)$ can be used in only one direction in a path that connects the terminal node u with the terminal node v. The connectivity requirements for each pair of terminal nodes are expressed by Equations (6)–(8). Equation (6) establishes that from the terminal node u at least $r_{u,v}$ disjoint paths must exist to the terminal node v. Equation (7) states that the terminal node v must reach at least $r_{u,v}$ disjoint paths from the terminal node u. Constraints in Equation (8) express the Kirchhoff flow conservation law, stating that the number of incident edges to an intermediate node p (denoted by the set $I\left(p\right)$) that are part of disjoint paths that lead from a terminal node u to a terminal node v is equal to the number of exit edges of the node p that are part of disjoint paths that lead from u to v. Constraints in Equation (9) state that any link incident to a Steiner node should only be used if the Steiner node is included in the solution, thus avoiding superfluous links that increase the network design cost. The constant value M is any large real number, for example $M=\left|E\right|$ without loss of generality. Constraint in Equation (10) establishes that the reliability of the solution subgraph $G_S\subseteq G$, built with all selected links $x_{i,j}$ must be greater or equal than the minimum reliability threshold $p_{min}$. In the equation $G_S\left(\left\{x_{i,j}\right\}\right)$ denotes that $G_S$ is the subgraph of G induced by the decision variables $\left\{x_{i,j}\right\}$ with value 1.

Finally, the set of constraints in Equations (11) and (12) indicate that all the considered decision variables are binary, i.e., included in $\{0,1\}$.

2.3. Generalized Steiner Problem with Node-Connectivity Constraints

The GSP-NC is a relaxation of the GSPNCHR. It is formally defined next.

Definition 2 (GSPNC).

GSPNC is the relaxation of GSPNCHR, without constraint (10). Specifically, given a simple undirected graph $G=(V,E)$, a set of distinguished nodes $T\subseteq V$ (called terminals), a matrix with symmetric link-costs ${\left\{c_{i,j}\right\}}_{(i,j)\in E}$ and a matrix of connectivity requirements $R={\left\{r_{i,j}\right\}}_{i,j\in T}$, build a minimum cost topology $G_S\subseteq G$ fulfilling the connectivity requirements.

In this work, a full algorithm for the GSP-NC is developed. Then, the number of feasible GSP-NC solutions that also meet the reliability threshold is studied. The key questions of the proposed research are strictly related with the number of feasible solutions for the GSP-NC that are also feasible for the GSPNCHR. This way, the proposed research studies the interplay between topological network design and reliability analysis. Furthermore, a sensitivity on the reliability parameters and connectivity constraints is also discussed.

2.4. NP-Hardness of GSP-NC and GSPNCHR Problem

The GSP-NC is NP-hard [15]. The considered GSPNCHR is also NP-hard, as is proved in this subsection.

Theorem 1.

The GSPNCHR is within the class of NP-hard problems.

Proof. 

The Hamilton Tour (i.e., Hamiltonian cycle problem) belongs to the list of 21 NP-complete problems proposed by Karp [16]. Consider a graph $G=(V,E)$ and the trivial instance of the GSPNCHR defined by $(G,C,R,T,P_E,P_{V-T},p_{min})$, with unitary costs, nodes and links that do not fail, all nodes are terminal, and two-path requirements $r_{i,j}=2$ for all $i,j\in V$. The network design cost is lower than the number of nodes $n=\left|V\right|$ if and only if the graph G has a Hamilton tour. Thus, even for the trivial instance, GSPNCHR is also NP-hard.    □

3. Related Work

This article extends the Generalized Steiner Problem (GSP), by considering connectivity requirements for nodes and a hostile reliability model. This section reviews related works about the analysis of network reliability, topological network design, and problems/models studying both topological and reliability properties.

Few articles have dealt with the topological optimization of network design jointly with reliability constraints. Barrera et al. [17] optimized the topological network design to minimize costs under reliability constraints (k-terminal). A Sample Average Approximation (SAA) method [18] was applied and authors concluded that suboptimal solutions are found if the model ignores dependent failures. Recent articles have studied reliability optimization in stochastic binary systems [19], and applying SAA methods [20]. A related and challenging research field is finding uniformly most reliable graphs, considering a given number of fixed nodes and links, with maximum uniform reliability [21].

A closer problem to the one addressed in this article is considering topological transformations replacing or changing links, or even paths or trees, with the main goal of increasing/improving the considered reliability estimation. A recent article [22] has introduced a new approach for building networks by applying transformation to increase reliability. The main theoretical results is that any graph with a cut-point is transformable into a bi-connected graph with an improved reliability. Considering this theoretical result, the proposed designs do not include cut-points.

Botton et al. [23] introduced the hop-constrained survivable network design problem with reliable edges. The problem proposes finding a minimum cost set of edges so that the induced subgraph contains at least k edge-disjoint paths containing at most L edges between each pair of nodes. Two variants of the problem were introduced: a static version where the reliability of edges is given, and an upgrading problem where edges can be upgraded to the reliable status at a given cost. Both approaches were modeled by mathematical programming and solved by Benders decomposition. Results showed that both variants are more difficult to solve than the original problem (without reliable edges) [24]. Results also proved the economic benefits of using the versions with reliability edges.

Ozcan et al. [25] proposed hybrid strategies to solve the minimum cost communication network design problem under all-terminal reliability constraint. Results showed that hybridizing metaheuristics with Branch-and-Bound is an effective approach to designing reliable networks and finding better solutions. Kabiri et al. [26] formulated the wireless backhaul network design problem under reliability and survivability constraints. An accelerated Benders decomposition algorithm was applied to solve the problem. Extensive computational experiments showed the accuracy of the designed model and the efficiency of the Benders method, particularly in larger networks. Liu et al. [27] proposed a reliable mixed integer programming optimization model for designing a minimum cost and survivable network that continues operation under node failures. Three heuristic methods were designed to solve the proposed models: Branch-and-Bound, global algorithm, and heuristic algorithm based on Benders decomposition. The computational results validated the accuracy of the proposed models and resolution methods.

Few works in the network reliability literature have considered dependencies between network components. The network reliability theory and the more general case of modeling generic systems prone to failures (i.e., stochastic binary systems) also assume the independence of component failures. Park [28] analyzed the exact time-dependent all-terminal reliability of wireless networks with dependent failures, to provide a more realistic assessment of the reliability. Hardware failures and channel fading in the static environment were considered. Exact time-dependent reliability polynomials were derived for two type of wireless networks: (i) regular networks where each node connects other nodes via four point-to-point radio modules, and (ii) random mesh networks where each node connects other nodes through redundant radio-broadcast modules. The exact instantaneous failure rates were computed too. Yang et al. [29] introduced a Bayesian reliability approach for complex systems with dependent life metrics. A method was proposed to convert the overall likelihood into a product of explicit and implicit evidence-based likelihood functions. The article developed a systematic analysis of the role of dependent evidence in reliability evaluation and the proposed full Bayesian approach was applied to various system reliability models. Moreover, different numerical cases and a practical engineering case were considered to demonstrate the validity of the proposed approach. A recent article by Cancela et al. [30] addressed the problem of estimating reliability for stochastic flow networks with dependent arcs. A splitting-based Monte Carlo method was proposed, using the Marshall–Olkin copula for the case of nonindependent components. The article introduced a review of the methods based on creation and the destruction processes. A comparative experimental analysis showed the efficiency of the proposed approach.

Many articles in the network reliability literature have dealt with reliability evaluation rather than with reliability maximization. Simulations allow achieving a reasonable trade-off between the accuracy of results and the computational cost of the calculations [16]. Approaches based on Monte Carlo simulation methods allow finding point-wise estimations via independent replications and statistical modeling of complex systems. The RVR method has proven to be a useful and precise technique for reliability estimation [12], significantly improving over the standard Monte Carlo approaches In this article, the RVR method is applied for estimating the network reliability. The application is meaningful since the RVR method has been extended to Stochastic Monotone Binary Systems [31], such as the hostile model studied in this article.

The GSP proposes finding a minimum cost network to communicate a given subset of terminal nodes, meeting connectivity requirements defined by paths that are disjoint on edges (GSP-EC variant) or disjoint on nodes (GSP-NC variant). The GSP is NP-hard, thus approximation algorithms and metaheuristics have been proposed as resolution approaches. Agrawal and Ravi [32] proposed a logarithmic factor approximation algorithm to solve the GSP-EC and later Jain [33] proposed a factor-2 approximation algorithm for the same problem, applying the primal-dual approach for linear programming. Suzuki et al. [34] ennumerated optimal GSP solutions applying the Zero-Suppressed Binary Decision Diagrams compact data structure. The proposed method worked correctly for a number of realistic problem instances. Heuristics and metaheuristics have also been proposed for the GSP. Sartor and Robledo applied a resolution methodology based on GRASP to solve the GSP-EC variant [35]. Nesmachnow [36] presented an empirical evaluation of metaheuristic algorithms (including evolutionary computation and VNS) for the GSP, with promising results for several problem instances.

4. Metaheuristic Approach for the GSPNCHR

This section presents the proposed metaheuristic approach applying GRASP and VND for solving the GSPNCHR.

4.1. Methodology

The following methodology is developed to solve the GSPNCHR. First, a GRASP/VND metaheuristic is applied to compute a complete GSP-NC of the problem without the reliability threshold (constraint (10) in the problem formulation). After that, the number of feasible solutions that satisfy constraint (10) is counted. The proposed approach relies on determining whether or not the topological robustness impacts on the reliability of the resulting network.

Since the GSP-NC is NP-hard, a metaheuristic approach is proposed to compute accurate solutions in reasonable execution times [37]. Two well-known metaheuristic approaches, namely GRASP and VND, are applied. These methods have been successfully applied to solve hard combinatorial optimization problems in the liteerature.

GRASP is a powerful multi-start process that operates in two phases [38]. In the first phase (the construction phase), a feasible solution is built. The second phase (local search phase) explores the neighborhood of the built solution. In turn, VND is a local search heuristic that explores several neighborhood structures. Unlike Variable Neighborhood Search (VNS), VND is based on exploring several neighborhood structures in a deterministic order [39]. The efficacy of VND relies on the fact that different neighborhood structures may have different local minima. The computed solution is, at the same time, a local optimum under all the considered neighborhood structures appllied in th search. This article develops a hybrid GRASP/VND metaheuristic, using VND in the local search phase of a GRASP. The general scheme of the proposed GRASP/VND approach is presented in the pseudocode in Algorithm 1.

Algorithm 1 Pseudocode of the proposed GRASP/VDN algorithm for the GSPNCHR

NetworkDesign$(G_B,iter,k,p_{min},P_E,P_{V-T},sim\_iter)$

1: $i\leftarrow 0$ 2: $P\leftarrow \varnothing$ 3: $sol\leftarrow \varnothing$ 4: while  $i<iter$  do 5:     ($\overline{g},P)\leftarrow ConstructionPhase(G_B,C,R,k,iter)$    ▹ Construction phase 6:     $g_{sol}\leftarrow VND(\overline{g},P)$                ▹ Local search/VND phase 7:     $reliability\leftarrow RVR(g_{sol},P_E,P_{V-T},sim\_iter)$    ▹ Reliability evaluation 8:     if $reliability>p_{min}$ then 9:         $sol\leftarrow sol\cup \left\{g_{sol}\right\}$ 10:     end if 11: end while 12: return  $sol$

The GRASP/VND algorithm receives seven input parameters:

• the ground graph $G_B$, including all possible connections to build the reliable topology
• a number of iterations $iter$, to be used as stopping criterion for the search
• a positive integer k, that determines the number of shortest paths to consider in the construction phase (see the description of the construction phase in the next paragraph)
• a reliability threshold $p_{min}$; it is a lower bound for the reliability of feasible network designs found by the algorithm;
• the elementary reliability $P_E$; a vector of operation probabilities for feasible links in set E;
• the elementary reliability $P_{V-T}$, a vector of operation probabilities for Steiner nodes in $V-T$;
• the number of iterations to be performed in the RVR simulations applied for reliability evaluation, $sim\_iter$.

The NetworkDesign algorithm works as follows. In lines 1–3, the parameters are initialized: the iteration counter i is set to zero), the set of node-disjoint paths P computed between pairs of terminals in T is set to empty, and the list $sol$ of feasible solutions that satisfy all the constraints of the GSPNCHR is set to empty too. The while loop on lines 4–11 iterates $iter$ times in the search for feasible solutions for the GSPNCHR and stores them in the $sol$ list. In line 5 the Construction algorithm builds a low-cost randomized greedy solution (based on the GRASP Construction Phase), which is stored in $\overline{g}$. In line 6, the designed VND algorithm is invoked, receiving as input the solution $\overline{g}$. VND performs a series of local searches looking for neighboring topologies with a better quality (i.e., lower cost and satisfying the connectivity requirements given by the matrix R) than the current solution, and returns the best neighbor topology found that fulfills the node-connectivity constraints R. The result topology is stored in $g_{sol}$. In line 8, the RVR method is applied to evaluate the reliability constraint, i.e., Equation (10) in the mathematical model. In case that the built solution $g_{sol}$ fulfills the reliability constraint (line 8), it is inserted in the set of solutions found, $sol$ (line 9). If $g_{sol}$ satisfies the condition of line 8 ($reliability$ > $p_{min}$), then it satisfies all the constraints of the GSPNCHR, and therefore the found topology $g_{sol}$ is a feasible solution of the GSPNCHR fulfilling both the node-connectivity constraints between pairs of terminal nodes as well as the reliability constraint, exceeding the pre-established reliability value $p_{min}$. Once the loop is finished, the list of low-cost feasible solutions found for the GSPNCHR is returned on line 12.

Next sections describe the construction phase and the local search/VND phase applied in the proposed algorithm.

4.2. Construction Phase

An iterative ad-hoc procedure is applied in the construction phase to build candidate solutions to be improved in the local search phase. The ad-hoc procedure was conceived to give a proper trade-off between simplicity and effectiveness. The ad-hoc construction procedure works by building paths iteratively. The pseudocode of the construction procedure is presented in Algorithm 2.

Algorithm 2 receives as input five parameters: the ground graph $G_B$, including all possible connections to build the reliable topology; the link costs matrix C; the matrix with connectivity requirements R; the number of shortest paths to compute between terminals k, and the number of iterations (max_iter). The set of terminal nodes is denoted $S_D^{\left(I\right)}$.

Algorithm 2 Pseudocode of the iterative ad-hoc procedure applied in the construction phase of the proposed GRASP/VDN algorithm for the GSPNCHR

procedureConstruction Phase$(G_B,C,R,k,max\_iter)$ 1: $g_{sol}\leftarrow (S_D^{\left(I\right)},\varnothing )$ 2: $m_{i,j}\leftarrow r_{i,j}$ 3: $P_{i,j}\leftarrow \varnothing ,\forall i,j\in S_D^{\left(I\right)}$ 4: $A_{i,j}\leftarrow 0,\forall i,j\in S_D^{\left(I\right)}$ 5: while$\exists m_{i,j}>0:A_{i,j}<$max_iterdo 6:     $(i,j)\leftarrow ChooseRandom(S_D^{\left(I\right)}:m_{i,j}>0)$ 7:     $\overline{G}\leftarrow G_B\backslash P_{i,j}$ 8:     for all $(u,v)\in E\left(\overline{G}\right)$ do 9:         ${\overline{c}}_{u,v}\leftarrow c_{u,v}\times 1_{\{(u,v)\notin g_{sol}\}}$ 10:     end for 11:     $L_p\leftarrow KSP(k,i,j,\overline{G},\overline{C})$ 12:     if $L_p=\varnothing$ then 13:         $A_{i,j}\leftarrow A_{i,j}+1$; $P_{i,j}\leftarrow \varnothing$; $m_{i,j}\leftarrow r_{i,j}$ 14:     else 15:         $p\leftarrow SelectRandom\left(L_p\right)$; $g_{sol}\leftarrow g_{sol}\cup \left\{p\right\}$ 16:         $P_{i,j}\leftarrow P_{i,j}\cup \left\{p\right\}$ 17:         $m_{i,j}\leftarrow m_{i,j}-1$ 18:         $(P,M)\leftarrow GeneralUpdateMatrix(g_{sol},P,M,p,i,j)$ 19:     end if 20: end while 21: return$(g_{sol},P)$

The initial solution $g_{sol}$ only contains the terminal nodes $S_D^I$, without including any link (line 1). Matrix $M={\left\{m_{i,j}\right\}}_{i,j\in T}$ is used to store requirements that are not satisfied yet, thus it in initialized as $m_{i,j}=r_{i,}$ for all $i,j\in S_D^{\left(I\right)}$ (line 2). Matrix $P={\left\{P_{i,j}\right\}}_{i,j\in S_D^{\left(I\right)}}$ represents the collection of node disjoint paths, which is initialized as empty for all $P_{i,j}$ (line 3). Matrix $A={\left\{A_{i,j}\right\}}_{i,j\in S_D^{\left(I\right)}}$ stores the number of iterations (or attempts) required by the algorithm to find $r_{i,j}$ node-disjoint paths between nodes $i,j$ is initialized as $A_{i,j}=0\forall i,j\in S_D^{\left(I\right)}$ (line 4).

The while loop in lines 5–20 iterates max_iter attempts to find paths that satisfy all the connectivity requirements. A large Restricted Candidate List is selected for diversification purposes. In each iteration, a pair of terminals $(i,j)$ satisfying $m_{i,j}>0$ is uniformly selected following a uniform distribution from the set $S_D^{\left(I\right)}$ (line 6). A new graph $\overline{G}$ is defined from the information in $G_B$, by discarding the nodes already visited in the previously computed paths (line 7). Any path between i and j in $\overline{G}$ is included in subset $P_{ij}$ of the set P. For the next paths to be computed between i and j, the nodes of $P_{ij}$ cannot be taken into account, in order to guarantee that the node-disjoint constraint between previously computed paths. The for loop in lines 8–10 builds an auxiliary matrix to store costs $\overline{C}=\left\{\overline{c_{i,j}}\right\}$. It keeps unchanged the costs of the edges that are not included in the matrix of paths between terminal nodes, and assigns cost zero to any edge that belongs to a path already computed and stored. This way, edges already included in the solution being constructed have zero cost in $\overline{C}$. Matrix $\overline{C}$ allows using links from $g_{sol}$ without increasing the cost. Line 11 computes the k shortest paths from i to j using Yen algorithm [40]. Then, it is tested if list $L_p$ is empty. In this case, $P_{i,j}$, $m_{i,j}$ is re-initialized and one unit is added to $A_{i,j}$. Otherwise, a path p is selected following a uniform distribution from the list $L_p$, to be included in the solution (line 15). The path p is added to $P_{i,j}$, and the corresponding requirement $m_{i,j}$ is decreased one unit (lines 16 and 17). Adding path p can define node-disjoint paths between terminal nodes. Thus, function GeneralUpdateMatrix is applied to find possible new paths (line 18). It updates the matrix M of connectivity deficits between pairs of terminals. For example, if the path p computed in line 15 contains a terminal node k, it is checked whether $p(i,k)$ and $p(j,k)$ (subpaths of p connecting nodes i and k, and k an j, respectively) are node-disjoint with respect to the sets $P_{ik}$ and $P_{kj}$. If either or both of them are disjoint, $m_{ik}$, $P_{ik}$ and/or $m_{kj}$, $P_{jk}$ are updated. This procedure is performed for every terminal node on p. In turn, if two terminals u and v in p are such that the intersection of $p_{u,v}$ and the paths at $P_{u,v}$ are node-disjoint, except for u and v, then $m_{u,v}$ is updated (decremented by 1) and the set $P_{u,v}$ is updated by adding the path $p(u,v)$.

The construction phase returns a feasible solution $g_{sol}$ including all the sets $P={\left\{P_{i,j}\right\}}_{i,j\in S_D^{\left(I\right)}}$ of node disjoint pairs between terminal nodes (line 21). The returned solution is feasible for the GSPNC.

4.3. Local Search (VND) Phase

The rationale behind the proposed VND is combining different neighborhoods to compute locally optimal solutions for each one of them. The proposed VND combines three neighborhood structures. Neighborhood are defined considering the concepts of key-nodes, key-paths, and key-trees, which are defined next.

Definition 3 (key-node).

A key-node in a feasible solution $v\in g_{sol}$ is a Steiner node with degree three or greater.

Definition 4 (key-path).

A key-path in a feasible solution $p\subseteq g_{sol}$ is an elementary path where all the intermediate nodes are Steiner nodes with degree 2 in $g_{sol}$, and the extremes are either terminals or key-nodes. A feasible solution $g_{sol}$ accepts a decomposition into key-paths, i.e., $K_{g_{sol}}=\{p_1,\dots ,p_h\}$.

Definition 5 (key-tree).

Consider a key-node v which is part of a feasible solution $g_{sol}$. $T_v$ is the key-tree associated to v, which is composed by all the key-paths with the common end-point v.

The three neighborhood structures that combine the previous concepts are defined next, considering a feasible solution $g_{sol}$ for the GSPNC.

Definition 6 (Neighborhood Structure for swap key-paths).

Given a key-path $p\subseteq g_{sol}$, a neighbor solution for $g_{sol}$ is defined by ${\widehat{g}}_{sol}=\{g_{sol}\backslash p\}\cup \left\{m\right\}$, being m the set of nodes and links to be added to preserve the feasibility of ${\widehat{g}}_{sol}$.

Definition 7 (Neighborhood Structure for key-paths).

Given a key-path $p\in g_{sol}$, a neighbor-solution is defined by ${\widehat{g}}_{sol}=\{g_{sol}\backslash p\}\cup \left\{\widehat{p}\right\}$, where $\widehat{p}$ is another path connecting the extreme nodes from p. The neighborhood of key-paths from $g_{sol}$ includes all feasible solutions computed by the previous operation on the key-paths in the key-path decomposition of $g_{sol}$ given by $K_{g_{sol}}$.

Definition 8 (Neighborhood Structure for key-tree).

Consider a key-tree $T_v\in g_{sol}$, rooted at key-node v. A neighbor of $g_{sol}$ is defined by ${\widehat{g}}_{sol}=\{g_{sol}\backslash T_v\}\cup \left\{T\right\}$, being T another tree that substitutes $T_v$, having identical leaf nodes.

A classical VND implementation is applied in the proposed GRASP/VND algorithm to solve the GSPNCHR. Local searches on the defined neighborhood structures are applied in order, after the construction phase:

• $SwapKeyPathLocalSearch$ (Algorithm 3).
• $KeyPathLocalSearch$ (Algorithm 4).
• $KeyTreeLocalSearch$ (Algorithm 5).

The order for applying the local search operators was evaluated in preliminary experiments. Even though the last two local search operators are simpler, the best results obtained in preliminary experiments suggested that $SwapKeyPathLocalSearch$ should be applied first.

Algorithm 3 Pseudocode of the SwapKeyPath local search procedure

procedureSwapKeyPathLocalSearch$(G_B,C,g_{sol})$ 1: $improvement\leftarrow TRUE$ 2: while$improvement$do 3:     $improvement\leftarrow FALSE$ 4:     $K\left(g_{sol}\right)\leftarrow \{p_1,\dots ,p_h\}$                                    ▹ Key-path decomposition of $g_{sol}$ 5:     while not $improvement$ and ∃ key-paths to be explored do 6:         $p\leftarrow \left(K\left(g_{sol}\right)\right)$                                                                ▹ Path not explored yet 7:         $(g_{sol},improvement)\leftarrow FindSubstituteKeyPath(g_{sol},p,P)$ 8:     end while 9: end while 10: return$g_{sol}$

Algorithm 4 Pseudocode of the KeyPath local search procedure

procedureKeyPathLocalSearch$(G_B,C,g_{sol})$ 1: $improvement\leftarrow TRUE$ 2: while$improvement$do 3:     $improvement\leftarrow FALSE$ 4:     $K\left(g_{sol}\right)\leftarrow \{p_1,\dots ,p_h\}$                                           ▹ Key-path decomposition of $g_{sol}$ 5:     while not $improvement$ and ∃ key-paths to be explored do 6:         $p\leftarrow \left(K\left(g_{sol}\right)\right)$                                         ▹ Path between u and v, not explored yet 7:         $\widehat{\mu }\leftarrow \langle NODES\left(p\right)\cup S_D\backslash NODES\left(g_{sol}\right)\rangle$                    ▹ Induced subgraph $\widehat{\mu }$ 8:         $\widehat{p}\leftarrow Dijkstra(u,v,\widehat{\mu })$ 9:         if $COST\left(\widehat{p}\right)<COST\left(p\right)$ then 10:            $g_{sol}\leftarrow \{g_{sol}\backslash p\}\cup \left\{\widehat{p}\right\}$ 11:            $improvement\leftarrow TRUE$ 12:         end if 13:     end while 14: end while 15: return$g_{sol}$

   The proposed local search operators are only applied to solutions that are feasible and have a lower cost than the original solution. Each local search follows the defined neighborhood structures, with the main goal of finding better solutions. Two functions are used in the searches:

• $FindSubstituteKeyPath$ (used in SwapKeyPath local search) receives the current solution $g_{sol}$, the key-path p and a matrix P with the collection of disjoint path between the terminals. The function replaces the current path p by $\widehat{p}$, exploiting the information given by P in order to reconstruct a new feasible solution. If the built solution is cheaper, the function returns the boleean value $improvement=1$ and the resulting solution. If a better solution is not found, $improvement=0$ and the current solution $g_{sol}$ is returned.
• $GeneralRecConnect$ searches for a key-tree T that spans the same leaf-nodes than key-tree $T_v$, improving (i.e., reducing) the cost and maintaining feasibility. If a new solution with a lower cost is found, the function returns the new solution and a boolean value $improvement=1$. If a better solution is not found, $improvement=0$ and the current solution $g_{sol}$ is returned.

Algorithm 5 Pseudocode of the KeyTreeLocal local search procedure

procedureKeyTreeLocalSearch$(G_B,C,g_{sol})$ 1: $improvement\leftarrow TRUE$ 2: while$improvement$do 3:     $improvement\leftarrow FALSE$ 4:     $X\leftarrow KeyNodes\left(g_{sol}\right)$                                                        ▹ Key-nodes from $g_{sol}$ 5:     $\overline{S}\leftarrow S_D\backslash NODES\left(g_{sol}\right)$ 6:     while not $improvement$ and ∃ key-nodes not explored do 7:         $v\leftarrow X$                                                                      ▹ Key-node not explored yet 8:         $(g_{sol},improvement)\leftarrow GeneralRecConnect(G_B,C,g_{sol},v,\overline{S})$ 9:     end while 10: end while 11: return  $g_{sol}$

4.4. Reliability Estimation

In the third phase, the RVR method is applied to estimate the reliability of the GSP-NC solution built using the GRASP/VND metaheuristic. The underlying probabilistic model is the hostile model that assumes probabilistic failures for links and Steiner nodes. Given the monotonicity of the hostile model, the application of RVR is suitable for this purpose [12,31,41]. RVR is efficient when applied to monotone models, i.e., models where a supra-state of an operational state is also an operational state. After computing an estimation of the reliability for each solution, it is determined if constraint (10) is met. If the reliability of the built network is greater than parameter $p_{min}$, the solution $g_{sol}$ is added to the set of reliable solutions $sol$.

5. Experimental Evaluation

  This section describes the experimental evaluation of the proposed GRASP/VND algorithm to solve the GSPNCHR.

5.1. Methodology

Two analysis were performed in the experimental validation of the proposed methods. First, the capabilities of the proposed GRASP/VND and reliability evaluation approach for the GSP-NC were evaluated. After that, a sensitivity analysis was performed to study the impact of perturbations in the elementary reliabilities on edges and Steiner nodes. Representative problem instances were considered in the analysis (two specific test sets were designed, see a description on the next subsection). The value of the lower bound for the reliability of the designed networks $p_{min}$ was set to 0.8 for all experiments.

In the GRASP algorithm, the size k of the candidate list used in the construction phase (i.e., list of low-cost paths between specific pairs of terminals) was set to 5. This parameterization was defined after performing a preliminary sensitivity analysis performed on random graphs that studied the impact of different values of k on the quality, particularly the cost, of the feasible solutions computed by GRASP/VND. Good quality (local) optimal feasible solutions were obtained in reasonable computational times (few seconds in most cases) for the GSPNC, using $k=5$.

The experimental validation was performed on the high performance computing platform of National Supercomputing Center (Cluster-UY), Uruguay [42].

5.2. Description of the Test Sets

This subsection describes the test sets used in the experimental evaluation of the proposed method to solve the GSPNCHR.

5.2.1. Overall Description

To the best of our knowledge, there are no benchmark instances or graph libraries, neither to evaluate the performance of exact and/or approximate techniques to solve the GSP or to evaluate network reliability. For reliability evaluation, classical graphs are generally used, e.g., dodecahedron, Arpanet, grids, some topologies of Halin and outerplanar graphs, and graphs depending on the base model (source-terminal reliability, k-terminal reliability, all-terminal reliability, etc.) [43,44]. Regarding the evaluation of custom algorithms to solve the GSP, the common approach in the literature is based on transforming instances from the the well-known TSPLIB library [45]. Therefore, two benchmark test sets were developed by adapting instances from TSPLIB: medium and large sized instances from different TSPLIB families were selected and transformed them into GSPNCHR instances following a specific methodology for test case generation.

Selected instances from TSPLIB were modified to get complete graphs, with the corresponding Euclidean costs on the links. The following instances were selected for the experimental evaluation of the proposed approach to solve the GSPNCHR: att48, berlin52, brazil58, ch150, d198, eil51, gr137, gr202, kroA100, kroA150, kroB100, kroB150, kroB200, lin105, pr152, rat195, st70, tsp225, u159, rd100, and rd400. The suffix of each name is the number of nodes in the corresponding instance (e.g., kroA100 has 100 nodes).

Two tests sets were defined, one for evaluating the GRASP/VND and reliability results and another for a sensitivity analysis to perturbations in the elementary reliabilities. These problem sets are described next.

5.2.2. Test Set for the GRASP/VND and Reliability Evaluation

Table 1. Description of the problem instances used for the evaluation of the proposed GRASP/VND and reliability results.

Instance Name	% T	${\mathit{p}}_{\mathit{v}}$–${\mathit{p}}_{\mathit{e}}$	%${\mathit{T}}_2$–${\mathit{T}}_3$–${\mathit{T}}_4$	iter_ND	iter_RVR	#
att48	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
berlin52	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
brazil58	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
ch150	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
d198	20–35–50	0.99–0.95	100–0–0	20–20–20	NA	3
eil51	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
gr137	20–35–50	0.99–0.95	100–0–0	100–20–20	NA	3
gr202	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
kroA100	20–35–50	0.99–0.95	100–0–0	100–100–100	NA	3
kroA150	20–35–50	0.99–0.95	100–0–0	100–20–20	NA	3
kroB100	20–35–50	0.99–0.95	100–0–0	100–100–100	NA	3
kroB150	20–35–50	0.99–0.95	100–0–0	100–20–20	NA	3
kroB200	20–35–50	0.99–0.95	100–0–0	20–20–20	NA	3
lin105	20–35–50	0.99–0.95	100–0–0	100–100–100	NA	3
pr152	20–35–50	0.99–0.95	100–0–0	20–20–20	NA	3
rat195	20–35–50	0.99–0.95	100–0–0	20–20–20	NA	3
st70	20–35–50	0.99–0.95	100–0–0	100–100–100	10000	3
tsp225	20–35–50	0.99–0.95	100–0–0	50–50–50	10000	3
u159	20–35–50	0.99–0.95	100–0–0	20–20–20	NA	3
rd100	20–35–50	0.99–0.95	100–0–0	100–100–100	NA	3
rd400	20–35–50	0.99–0.95	100–0–0	50–50–50	10000	3
berlin52(E)	20	0.99–0.90	65–25–10	100	10000	1
eil51(E)	20	0.99–0.90	65–25–10	100	10000	1
att48(E)	35	0.99–0.90	65–25–10	100	10000	1
st70(E)	35	0.99–0.90	65–25–10	100	10000	1
brazil58(E)	50	0.99–0.90	65–25–10	100	10000	1
eil51(E)	50	0.99–0.90	65–25–10	100	10000	1
kroB100(E)	20	0.99–0.90	65–25–10	100	10000	1
lin105(E)	20	0.99–0.90	65–25–10	100	NA	1
kroA100(E)	35	0.99–0.90	65–25–10	20	10000	1
rd100(E)	35	0.99–0.90	65–25–10	20	NA	1

describes the main features of the considered problem instances. The table reports for each instance the following information:

• instance name: the name of the instance. A name including (E) means that the instance is a variation of the corresponding base instance, with different connectivity requirements.
• %T: the percentage of terminal nodes in the graph. The following values were considered to build the test set: 20%, 35% y 50%. Thus, three GSP-NC instances are defined for each TSP instance.
• $p_v$–$p_e$: the elementary reliabilities for Steiner nodes ($p_v$) and links ($p_e$).
• %$T_2$–$T_3$–$T_4$: the percentage of pairs (of terminal nodes) that must meet connectivity requirements $r_{i,j}\in \{2,3,4\}$, respectively.
• $iter\_ND$: number of iterations considered in the NetworkDesign procedure, defined according to %T.
• $iter\_RVR$: number of iterations considered in $RVR$ method.
• #: number of generated instances.

A total number of 73 problem instances were defined. All of them were applied for the evaluation of the proposed GRASP/VND metaheuristic approach to solve the GSPNC. In turn, a subset of 38 problem instances were applied to evaluate the constraint imposed by the reliability threshold.

The maximum iterations for the NetworkDesign procedure was set to $iter\_ND=100$ for those instances with relative small CPU times (minutes), and $iter\_ND\in \{20,50\}$ for instances demanding larger CPU times. The number of iterations for the RVR method was set to $iter\_RVR={10}^4$, after results obtained in preliminary calibration experiments. In those instances not used for the reliability analysis, the abbreviation NA (for not applicable) is reported.

5.2.3. Test Set for the Sensitivity Analysis to Perturbations in the Elementary Reliabilities

A specific test-set was designed for experiments oriented to answer the strategic questions of the research (RQ 1–RQ 4), by studying the sensitivity of the computed solution to perturbations in the elementary reliabilities. In this dataset, different values were considered for the elementary reliabilities of Steiner nodes ($p_v$) and links ($p_e$). Specifically, the nine combinations of $p_v$ and $p_e\in \{0.99,0.97,0.95\}$ were introduced in different instances from the GRASP/VND and reliability evaluation test set. The details of the sensititivity analysis test set are presented in

Table 2. Description of the problem instances used for the sensitivity analysis to perturbations in the elementary reliabilities.

Instance Name	% T	%${\mathit{T}}_2$–${\mathit{T}}_3$–${\mathit{T}}_4$	iter_ND	iter_RVR	#
att48	20–35–50	100–0–0	100	10000	3
att48(E)	20	0-100-0	100	10000	1
att48(E)	20	0-0-100	100	10000	1
att48(E)	35	65–25–10	100	10000	1
berlin52	20–35–50	100–0–0	100	10000	3
berlin52(E)	20	65–25–10	100	10000	1
brazil58	20–35–50	100–0–0	100	10000	3
brazil58(E)	50	65–25–10	100	10000	1
eil51	20–35–50	100–0–0	100	10000	3
eil51(E)	20	65–25–10	100	10000	1
eil51(E)	50	65–25–10	100	10000	1
kroA100	35	100–0–0	100	10000	1
kroA100(E)	35	65–25–10	100	10000	1
kroB100	20	100–0–0	100	10000	1
kroB100(E)	20	65-20-10	100	10000	1
ch150	20–35–50	100–0–0	100	10000	3
gr202	20–35–50	100–0–0	100	10000	3
tsp225	20–35–50	100–0–0	100	10000	3
rd400	20–35–50	100–0–0	100	10000	3

.

Table 2 describes the generated instances. For instance, the first row indicates that three instances were generated using problem instance att48, with 20%, 35%, and 50% of terminal nodes, respectively, and connectivity requirements $r_{u,v}=2$ (100–0–0). For each instance based on att48, one-hundred feasible solutions were found, using 10,000 iterations of the RVR method considering the nine possible scenarios of elementary reliabilities ($p_v$,$p_e$): (0.99, 0.99), (0.99, 0.97), (0.99, 0.95), (0.97, 0.99), (0.97, 0.97), (0.97, 0.95), (0.95, 0.99), (0.95, 0.97), and $(0.95,0.95)$.

5.3. Numerical Results

This subsection reports the numerical results of the proposed approach for solving the GSPNCHR.

5.3.1. GRASP/VND and Reliability Evaluation

Table 3. Numerical results of the proposed GRASP/VND metaheuristic.

Instance Name	% T	% $\mathbf{IG}$	% IVND	$\mathbf{CPU}$ (s)	$\overline{\mathit{R}}$	$\overline{\mathbf{Var}}$
att48	20	99.27	34.61	11.46	0.967	7.60 × 10${}^{-7}$
att48	35	98.60	36.83	29.76	0.943	3.44 × 10${}^{-6}$
att48	50	98.22	37.10	65.90	0.927	5.32 × 10${}^{-6}$
berlin52	20	98.98	30.55	30.60	0.937	3.294 × 10${}^{-6}$
berlin52	35	99.06	33.93	33.43	0.938	3.19 × 10${}^{-6}$
berlin52	50	98.02	33.48	106.94	0.907	6.487 × 10${}^{-6}$
brazil58	20	98.92	31.96	62.37	0.885	6.722 × 10${}^{-6}$
brazil58	35	99.25	39.45	68.89	0.860	8.34 × 10${}^{-6}$
brazil58	50	98.75	35.26	103.55	0.910	7.09 × 10${}^{-6}$
ch150	20	99.76	37.51	222.55	0.856	1.02 × 10${}^{-5}$
ch150	35	99.72	36.65	546.65	0.880	9.03 × 10${}^{-5}$
ch150	50	99.69	34.42	1203.05	0.888	8.97 × 10${}^{-5}$
d198	20	99.90	32.22	320.14	NA	NA
d198	35	99.86	34.12	2086.37	NA	NA
d198	50	99.81	33.39	5548.63	NA	NA
eil51	20	99.34	38.79	14.87	0.960	1.18 × 10${}^{-6}$
eil51	35	98.54	36.11	39.02	0.942	3.73 × 10${}^{-6}$
eil51	50	98.56	37.32	44.79	0.937	4.28 × 10${}^{-6}$
gr137	20	99.79	36.31	137.49	NA	NA
gr137	35	99.71	34.18	404.06	NA	NA
gr137	50	99.68	34.61	976.36	NA	NA
gr202	20	99.89	32.43	528.16	0.823	1.22 × 10${}^{-5}$
gr202	35	99.75	34.56	3511.69	0.841	1.11 × 10${}^{-5}$
gr202	50	99.74	33.36	9505.62	0.830	1.27 × 10${}^{-5}$
kroA100	20	99.61	36.77	44.22	NA	NA
kroA100	35	99.53	38.23	101.49	0.890	8.52 × 10${}^{-5}$
kroA100	50	99.45	35.89	280.83	NA	NA
kroA150	20	99.83	36.70	102.71	NA	NA
kroA150	35	99.75	36.30	412.97	NA	NA
kroA150	50	99.70	32.32	2035.06	NA	NA
kroB100	20	99.68	38.71	17.30	0.901	6.25 × 10${}^{-5}$
kroB100	35	99.59	36.32	53.74	NA	NA
kroB100	50	99.49	34.98	191.72	NA	NA
kroB150	20	99.84	37.49	112.09	NA	NA
kroB150	35	99.77	36.05	665.67	NA	NA
kroB150	50	99.73	34.53	1327.52	NA	NA
kroB200	20	99.89	36.14	279.15	NA	NA
kroB200	35	99.84	35.06	2234.73	NA	NA
kroB200	50	99.8	33.82	7448.42	NA	NA
lin105	20	99.74	35.89	9.44	NA	NA
lin105	35	99.61	37.04	86.85	NA	NA
lin105	50	99.5	36.40	245.24	NA	NA
pr152	20	99.79	37.14	281.16	NA	NA
pr152	35	99.77	36.86	808.47	NA	NA
pr152	50	99.74	36.88	1673.46	NA	NA
rat195	20	99.88	37.31	280.94	NA	NA
rat195	35	99.82	34.70	1925.98	NA	NA
rat195	50	99.8	34.99	4599.87	NA	NA
st70	20	99.44	39.84	39.85	0.919	4.07 × 10${}^{-6}$
st70	35	99.3	39.56	63.65	0.906	5.74 × 10${}^{-6}$
st70	50	99.16	36.37	128.19	0.913	7.02 × 10${}^{-6}$
tsp225	20	99.88	34.98	1658.77	0.848	1.14 × 10${}^{-5}$
tsp225	35	99.85	34.65	4684.36	0.846	1.24 × 10${}^{-5}$
tsp225	50	99.82	33.26	12088.72	0.872	1.09 × 10${}^{-5}$
u159	20	99.81	35.84	333.26	NA	NA
u159	35	99.76	36.14	864.99	NA	NA
u159	50	99.75	35.61	1278.13	NA	NA
rd100	20	99.68	37.15	22.42	NA	NA
rd100	35	99.5	34.54	126.82	NA	NA
rd100	50	99.42	36.13	245.82	NA	NA
rd400	20	99.94	35.84	5000.21	0.809	14.22 × 10${}^{-5}$
rd400	35	99.94	33.54	9000.10	0.854	11.89 × 10${}^{-5}$
rd400	50	99.93	33.16	19000.70	0.864	11.51 × 10${}^{-5}$
berlin52(E)	20	98.45	25.25	34.20	0.993	4.84 × 10${}^{-7}$
eil51(E)	20	98.47	28.45	29.62	0.996	2.71 × 10${}^{-7}$
att48(E)	35	97.45	31.74	62.96	0.994	4.93 × 10${}^{-7}$
st70(E)	35	98.52	31.87	135.50	0.993	6.55 × 10${}^{-7}$
brazil58(E)	50	97.48	31.84	172.63	0.994	4.82 × 10${}^{-7}$
eil51(E)	50	97.26	32.67	74.47	0.991	7.94 × 10${}^{-7}$
kroB100(E)	20	99.37	30.25	39.25	0.999	1.21 × 10${}^{-6}$
lin105(E)	20	99.33	31.95	64.41	NA	NA
kroA100(E)	35	98.99	35.88	225.50	0.998	1.82 × 10${}^{-6}$
rd100(E)	35	99.15	35.30	130.01	NA	NA
average	-	99.39	35.03	1026.00	0.914	2.33 × 10${}^{-5}$

reports the results of the GRASP/VND metaheuristic for each problem instance in the evaluation test set. The first two columns indicate the name of the problem instance and the percentage of terminal nodes. Regarding numerical results, the following values are reported:

• % $IG$: percentage of cost reduction of the solution generated by the GRASP construction phase with respect to the input graph $G_B$, which contains all the feasible links between pairs of nodes.
• % $IVND$: percentage of improvement of the VND search over the cost of the iterative construction phase.
• $CPU$: average CPU-time per iteration of NetworkDesign.
• $\overline{R}$: average of the reliability estimation.
• $\overline{Var}$: average of the estimated variance associated to the reliability estimator of RVR.

An important aspect when analyzing the results of hybrid algorithms is to determine the ability of each component to improve over the baseline values. Results in Table 3 demonstrate that the relative improvement of VND over the cost of the solution built in the construction phase was between 25.25% and 39.84%. The average improvement over all considered problem instances was 35.03%, a significant value that demonstrate the efficacy of the proposed VND search. In turn, the minimum threshold $p_{min}=0.8$ was significantly exceeded in all problem instances where the average reliability $\overline{R}$ was estimated. For those instances in which the elementary reliabilities were established in 0.99–0.95 for nodes and links, respectively, the resulting reliabilities of the computed solutions were between 82.31% and 96.7%. When considering elementary reliabilities of 0.99–0.90, the resulting reliabilities of the computed solutions were between 99.1% and 99.87%.

Sample graphical results are presented for two representative problem instances in the test set (instances Brazil58 and Berlin52). For each instance, the ground topology is presented and the corresponding network design found by the proposed GRASP/VND metaheuristic is shown. Terminal nodes are represented as red circles and Steiner nodes are represented as orange circles.

Both sample problem instances include at least 20% of terminal nodes, the elementary reliability in Steiner nodes was set to 0.99, the elementary reliability for links was set to 0.95 and the connectivity requirements demand two node-disjoint paths between each pair of terminal nodes.

Figure 1 and Figure 2 presents the results for problem instance Brazil58. The resulting cost was 25,106 and the computed reliability was 0.9174.

Figure 3 and Figure 4 presents the results for problem instance Berlin52. The resulting cost was 4534 and the computed reliability was 0.8448.

Sample results in Figure 1, Figure 2, Figure 3 and Figure 4 show that the proposed GRASP/VND algorithm was able to compute simple, yet accurate network designs for the considered problem instances.

The execution time significantly increased for the most dense topologies solved, as the percentage of terminal nodes increased. On the one hand, instances with 20% of terminal nodes were solved in less than five minutes for all topologies. On the other hand, some instances with 50% of terminal nodes demanded more than 10.000 seconds, i.e., more than three hours of execution time. These efficiency results are coherent with the complexity of the problem, which is related to the dimension of the set of terminal nodes.

The estimated variance was reduced in average in all the studied instances. This result suggests that the RVR method is accurate, even under reliability failures of $q={10}^{-2}$ for both Steiner nodes and links. This fact is further analyzed in the sensitivity analysis presented in the following subsection.

5.3.2. Sensitivity Analysis to Perturbations in the Elementary Reliabilities

This subsection reports the results of the sensitivity analysis of the computed solution to perturbations in the elementary reliabilities. The main results and findings are discussed an related to the main research questions RQ 1–RQ 4 relating the network optimization and reliability evaluation, as defined in the introduction section.

Regarding RQ 1,

Table 4. Percentage of feasible GSPNCHR solutions verifying $R\ge 0.98$.

Instance	%T	Solutions with $\mathit{R}\ge 0.98$	Instance	%T	Solutions with $\mathit{R}\ge 0.98$
att48	20	100%	ch150	20	100%
att48	35	100%	ch150	35	100%
att48	50	100%	ch150	50	100%
eil51	20	100%	gr202	20	99%
eil51	35	100%	gr202	35	100%
eil51	50	100%	gr202	50	100%
berlin52	20	100%	tsp225	20	100%
berlin52	35	100%	tsp225	35	100%
berlin52	50	100%	tsp225	50	100%
brazil58	20	99%	rd400	20	100%
brazil58	35	97%	rd400	35	100%
brazil58	50	100%	rd400	50	100%

reports the number of feasible network designs found by the proposed GRASP/VND metaheuristic with a high estimated network reliability value (i.e., $R>p_{min}=0.98$).

In Table 4, the reliability was estimated for an scenario that assumes specific values in the underlying probabilistic failure model: elementary reliabilities in Steiner nodes and links were defined at 0.99. Three variants of each problem instance were considered, varying the number of terminal nodes to be 20%, 35%, and 50% of the total nodes.

   The information reported in Table 4 provides an insight of the level of feasibility computed by the GRASP/VND algorithm, providing a global answer to RQ 1. The number of computed solutions that met the reliability threshold was high. Indeed, 100% of the returned solutions satisfied the reliability threshold in all but three problem instances.

Regarding RQ 2, the sensitivity analysis studied the impact on the estimated reliability of the resulting network design in two cases: (i) fixing the elementary reliability value for Steiner nodes to $p_v=0.99$ and considering different values for the elementary link reliabilities $p_e\in \{0.99,0.97,0.95\}$; and (ii) fixing the elementary reliability value for links to $p_e=0.99$ and considering different values for the elementary reliabilities of Steiner nodes $p_v\in \{0.99,0.97,0.95\}$. In both cases, the number of feasible networks that survive, according to the defined reliability threshold (i.e., $R>p_{min}=0.98$), is evaluated.

Table 5. Solutions such that $R>p_{min}=0.98$. for $p_e$ fixed to 0.99 (left) and for $p_v$ fixed to 0.99 (right).

Instance	Fixed Elementary Reliability for Links				Fixed Elementary Reliability for Nodes
	0.99–0.99	0.99–0.97	0.99–0.95		0.99–0.99	0.97–0.99	0.95–0.99
att48 T20	100	90	12		100	100	99
att48 T35	100	53	0		100	98	96
att48 T50	100	20	0		100	100	99
berlin52 T20	100	41	0		100	100	80
berlin52 T35	100	50	0		100	99	93
berlin52 T50	100	1	0		100	100	100
brazil58 T20	99	15	0		99	59	41
brazil58 T35	97	0	0		97	43	9
brazil58 T50	100	5	0		100	99	81
ch150 T20	100	0	0		100	60	20
ch150 T35	100	0	0		100	98	76
ch150 T50	100	0	0		100	100	97
gr202 T20	99	0	0		99	80	30
gr202 T35	100	0	0		100	69	16
gr202 T50	100	0	0		100	100	76
rd400 T20	100	0	0		100	16	2
rd400 T35	100	0	0		100	98	80
rd400 T50	100	0	0		100	100	100

analyzes the number of feasible solution designs for each case. The table reports the percentage of feasible solutions for the GSPNCHR (reliability higher than $p_{min}$) computed by the proposed GRASP/VND metaheuristic, over the total number of independent executions of the NetworkDesign algorithm.   

   Results in Table 5 demonstrate that two different situations arise from varying the elementary reliabilities of links and Steiner nodes. On one hand, fixing the elementary reliability for nodes and reducing the elementary reliabilities for links, a notorious reduction in the number of feasible solutions is observed, even reaching the extreme value $0\%$ for all but one problem instance when the link reliabilities are 0.95. For probabilities of operation 0.99 for Steiner nodes and 0.97 for links, the number of feasible solutions of the GSPNCHR found was zero for networks with 150 nodes or more.

On the other hand, fixing the link reliabilities and varying the node reliabilities, the reduction of the resulting network reliability is less important than in the previous case, in terms of the percentage of feasible solutions found by the NetworkDesign algorithm for the GSPNCH. Results show that when the number of terminal nodes decreased, the corresponding reduction in the percentage of feasible solutions found for the GSPNCHR was also reduced, as the probabilities of operation exposed to the links decreased. Given an instance with a certain number of terminal nodes, the reduction in the number of feasible solutions found for the GSPNCHR is smaller for the case where the operation probabilities for Steiner nodes are fixed and the link operation probabilities decrease (columns 2 to 4 in Table 5) with respect to the case where the link operation probabilities are fixed and the Steiner node operation probabilities decrease (columns 5 to 7 in Table 5).

Regarding RQ 3, the main goal was analyzing the sensitivity of the model with respect to the connectivity constraints defined for terminal nodes. More precisely, given a triplet $(p_{min},P_E,P_{V-T})$ and considering a given baseline network topology, different input networks are generated for the GSPNCHR, considering different number of terminal nodes and different connectivity requirements for each pair of terminal nodes. For those instances, the number of solutions computed by the proposed GRASP/VND metaheuristic whose reliability is over $p_{min}$ is analyzed. The considered connectivity requirements for pairs of terminal nodes were $r_{i,j}\in \{2,3,4\}$, $i,j\in T$.

Table 6. Solutions such that $R\ge 0.98$ (case 0.99–0.97).

Instance	%T	%${\mathit{T}}_2$–${\mathit{T}}_3$–${\mathit{T}}_4$	Feasible Solutions with $\mathit{R}\ge 0.98$
att48	20	(100–0–0)	90%
att48	20	(65–25–10)	100%
att48	20	(0-100-0)	100%
att48	20	(0-0-100)	100%
eil51	20	(100–0–0)	76%
eil51	20%	(65–25–10)	100%
eil51	50%	(100–0–0)	54%
eil51	50%	(65–25–10)	100%
berlin52	20%	(100–0–0)	41%
berlin52	20%	(65–25–10)	100%
brazil58	50%	(100–0–0)	5%
brazil58	50%	(65–25–10)	100%
kroA100	35%	(100–0–0)	0%
kroA100	35%	(65–25–10)	100%
kroB100	20%	(100–0–0)	3%
kroB100	20%	(65–25–10)	100%

reports sample results obtained for problems with the parameter setting $p_{min}=0.98$. The convention for the number of requirements defined in Section 5.2.2 is applied for $r_{u,v}$: %$T_2$–$T_3$–$T_4$ is the percentage of pairs (of terminal nodes) that must meet connectivity requirements $r_{i,j}\in \{2,3,4\}$.

Results in Table 6 indicate that increasing the network connectivity requirements always imply a corresponding increase in the percentage of networks that meet the reliability threshold, and vice-versa. Results demonstrate the interaction between the topological network design and the network reliability. In general, networks with high number of disjoint paths imposed as node-connectivity between pairs of terminal nodes are also robust networks from the point of view of their reliability (under the probabilistic model, where the elementary components of the network have an associated probability of operation), i.e., their reliability levels are high. Conversely, under probabilistic models where edges and/or nodes have associated operation probabilities, if a particular network topology has a high level of reliability (either in edge reliability, node reliability or hostile model) then, in general, it also implies the existence of redundancy in the network, either through multiple edge-disjoint paths between pairs of terminal nodes and/or multiple node-disjoint paths between pairs of terminal nodes.

In practice, in the design of WAN networks, for example, the level of 2-node-connectivity between nodes of greatest interest (modeled by terminals in the presented problem) is normally set, and the network is designed to meet this requirement at the lowest possible cost. Moreover, 2-connected networks (edge-connected or node-connected) have high levels of reliability when their elementary components are also highly reliable.

Regarding RQ 4, the main conclusion from results reported in Table 5 and Table 6 is that, in order to design a more robust network, the best decision is improving the probabilities of operation for links. The impact of elementary probabilities of links is significantly higher (better) than the impact of elementary probabilities of nodes.

Figure 5 summarizes the effect of variations on the elementary reliability values for links and nodes. The bar graph shows the overall average reliability of the constructed network designs reported in Table 5, considering fixed values for the elementary reliabilities of edges ($p_e=0.99$, blue bars) and nodes ($p_v=0.99$, orange bars). The varying elementary reliabilities in each case are represented for the lables $p*$ in the x axis.

The graphic in Figure 5 clearly shows that the effect of reducing the elementary reliabilities of edges is significantly more important than varying the elementary reliabilities of nodes. In fact, a small reduction of 0.01 on the elementary reliabilities of edges (from 0.99 to 0.98) reduced the reliability of the resulting designs from 99.72 to only 15.28, whereas a second reduction of 0.01 caused a negligible value of 0.68 for the reliability of the network. Reducing the elementary reliabilities of nodes only decreased the reliability of the network from 99.72 to 84.39 (reduction of 0.01) and to 66.39 (reduction of 0.02).

6. Special Cases of the GSPNCHR

This section presents particular cases of the GSPNCHR that can be solved in polynomial time. Theoretical results related to obtaining global optimal solutions in polynomial times are introduced. In turn, polynomial algorithms to solve these special cases are proposed. In particular, special cases linked to source-terminal reliability ($T=\{s,t\}$) and all-terminal reliability ($T=V$) scenarios are introduced. The special cases studied in this section are relevant, because they model relevant problems in practice, in different contexts.

6.1. Source-Terminal Reliability

Let $T=\{s,t\}$ be the terminal-set (this scenario is known as the source-terminal model) with $r_{s,t}=1$. Let l be the length of the shortest path $P_{s,t}$ connecting s and t, Lemma 1 holds.

Lemma 1.

The threshold $p_{min}$ is met with the shortest path if and only if $p^l\ge p_{min}$.

Proof. 

A globally optimum solution is met with the shortest path $G_S=P_{opt}$ under identical costs if l satisfies the previously stated inequality. The shortest path $P_{opt}$ is found using Dijkstra algorithm [46].    □

Proposition 1 relates the source-terminal scenario with the GSPNCHR.

Proposition 1.

If there exists a feasible single path $P_{s,t}$ for the source-terminal scenario with identical probabilities $p_{i,j}=p$, then the globally optimum solution for GSPNCHR can be found.

Proof. 

By hypothesis, a feasible path $P_{s,t}$ that meets the reliability threshold $p_{min}$ exists. In particular, the threshold is met by the shortest path. By Lemma 1, $p^l\ge p_{min}$. Consider the largest integer h such that $p^h\ge p_{min}$, defined by Equation (13).

$$h=\lfloor \frac{log\left(p_{min}\right)}{log\left(p\right)}\rfloor$$

(13)

The globally optimum solution for the GSPNCHR is obtained applying the Cheng-Ansari algorithm [47], finding the minimum cost among all the paths with lengths $i\in \{l,\dots ,h\}$.    □

Proposition 2.

For the source-terminal scenario, $T=\{s,t\}$, $P_E={\left\{p_{ij}\right\}}_{(i,j)\in E}$, $P_{V\backslash T}={\left\{1\right\}}_{v\in V\backslash T}$, $r_{st}=1$, $c_{ij}=c,\forall (i,j)\in E$, and a given $p_{min}$, the global optimal solution of the $GSPNCHR$ can be computed in polynomial time.

Proof. 

Given a path p communicating s and t in G, the reliability condition for this path is given by Equation (14).

$$\prod _{(i,j)\in p}{p}_{ij}\ge p_{min}.$$

(14)

The constraint is established by Equation (15), applying logarithm to both sides of the inequality.

$$\sum _{(i,j)\in p}(-log\left(p_{ij}\right))\le -log\left(p_{min}\right)$$

(15)

Consider the matrix $\widehat{P}={\{-log\left(p_{ij}\right)\}}_{(i,j)\in E}$. The length-bounded Dijsktra Algorithm, which computes the shortest path between two nodes with the condition that the path has no more than l edges (hops), a pre-established parameter, is applied until the first path that satisfies the reliability condition is found. If the GSPNCHR has a feasible solution, Algorithm 6 returns the global optimum for this particular case.    □

Algorithm 6 Pseudocode of GSP-NCHR Source Terminal reliability

procedureGSP-NCHRSourceTerminal ($G=(V,E)$, $T=\{s,t\}$, $\widehat{P}$, c, $p_{min}$) 1: $l\leftarrow 1$; 2: found_solution← FALSE; 3: while ($l\le \left|E\right|-1$) and not found_solution do do 4:     $(\widehat{p},cost)\leftarrow$ RestrictedDijkstra$(G,\widehat{P},s,t,l)$          ▹ Compute the bounded shortest path from                             ▹s to t based in matrix $\widehat{P}$. $\widehat{p}$ has no more than l hops 5:     if $cost\le -log\left(p_{min}\right)$ then 6:         found_solution← TRUE; 7:         optimal_solution $\leftarrow \widehat{p}$; 8:         optimal_solution $\leftarrow l·c$; 9:     else 10:         $l\leftarrow l+1$; 11:     end if 12: end while 13: iffound_solutionthen 14:     return (optimal_solution,optimal_cost); 15: end if

   Consider now the particular case defined by $T=\{s,t\}$, $r_{s,t}=k>1$. If the costs and the probabilities (in edges or nodes) are not uniform, the problem is NP-hard.

Two relevant sub-cases are studied:

(i)

The sub-case with non-uniform costs in the edges, uniform edge probabilities, and uniform node probabilities (i.e., $P_{V\backslash T}={\left\{1\right\}}_{v\in V\backslash T}$).

(ii)

The sub-case with uniform costs in the edges, non-uniform edge probabilities, and uniform node probabilities.

For both sub-cases, the Suurballe Algorithm [48] can be applied to determine if there are feasible solutions, as described in Proposition 3. The Suurballe algorithm computes in polynomial time the k-node-disjoint paths (or the k-edge-disjoint paths) from s to t of minimum total edge cost, for a given integer k.

Proposition 3.

[Existence of feasible solutions for sub-cases $\left(i\right)$ and $\left(ii\right)$] Lets graph $\widehat{G}$ be equal to G but such that each edge $e\in E$ is weighted by $-log\left(p_e\right)$. Lets denote $\widehat{C}={\{-log\left(p_e\right)\}}_{e\in E}$. Let $L={\left\{l_i\right\}}_{i\in 1\dots k}$ be the k-node-disjoint paths (respectively, k-edge-disjoint paths) with minimum sum of costs over $\widehat{G}$ between s and t considering $\widehat{C}$. Then, there is a feasible solution if the inequality given by Equation (16) holds.

$$\sum _{i=1}^k\sum _{e\in l_i}-log\left(p_e\right)\le -log\left(p_{min}\right)$$

(16)

Proof. 

If $L={\left\{l_i\right\}}_{i\in 1\dots k}$ were not feasible, the inequality ${\prod }_{i=1}^k{\prod }_{e\in l_i}{p}_e<p_{min}$ would hold, which would imply ${\sum }_{i=1}^k{\sum }_{e\in l_i}log\left(p_e\right)<log\left(p_{min}\right)$.    □

For the sub-case (i), the Suurballe Algorithm must be applied in combination with the Grötschel Algorithm when calculating a path that interlinks the remaining r paths.

For the sub-case (ii), since the costs of the edges are uniform, say equal to $c_0$, then the maximum possible cost is bounded by $m·c_0$, where m is the number of the edges in the graph. So, it is enough to find the k minimum logprob path (${\sum }_{e\in L_k}log\left(p_e\right)>log\left(p_{min}\right)$) with lower cost than $i·c_0$, with $i=1\dots m$. Algorithm 7 describes the proposed procedure for sub-case (ii) (i.e., $r_{s,t}=k>1$, and edges with uniform costs).   

Algorithm 7 Pseudocode of GSP-NCHR Source Terminal reliability, case (ii)

procedureGSP-NCHRSourceTerminal-k-conn ($\widehat{G}$, $\widehat{C}$,$T=\{s,t\}$, $P_E$, m, $p_{min}$) 1: $l\leftarrow 1$ 2: optimum← FALSE; 3: repeat 4:     $i\leftarrow i+1$ 5:     $L_k\leftarrow$ the k paths given by Suurballe$(\widehat{G},\widehat{C},k,s,t,i)$ using Grotschel; 6:     if $costo\left(L_k\right)\le -log\left(p_{min}\right)$ then 7:         optimum$\leftarrow \mathrm{TRUE}$ 8:     end if 9: untiloptimum or $i=m$ 10: ifoptimumthen 11:     return $L_k$ 12: else 13:     return no solution 14: end if

6.2. All-Terminal Reliability

Under the all-terminal reliability model, all the nodes are considered as terminal, i.e., $T=V$, and no Steiner (optional) nodes are taken into account. Consider the (cheapest) Minimum Spanning Tree $G_S=\mathcal{T}$. If $\mathcal{T}$ fulfills the reliability threshold, then the globally optimum solution is met.

Proposition 4 relates the all-terminal reliability model with the GSPNCHR.

Proposition 4.

Under the all-terminal reliability model, a Minimum Spanning Tree $\mathcal{T}$ achieves the globally optimum solution if and only if ${\prod }_{e\in \mathcal{T}}{p}_e\ge p_{min}$.

Proof. 

Under the all-terminal reliability model $T=V$, so the minimal feasible solutions for the GSPNCHR are spanning trees. Let $\mathcal{T}$ be a minumum spanning tree. The topology $\mathcal{T}$ is globally optimal if and only if it is globally optimal for the integer linear programming problem presented in Section 2.2. This condition is fulfilled since:

(i)

$\mathcal{T}$ minimizes Equation (3) (the objective function), since it is a minimum spanning tree.

(ii)

$\mathcal{T}$ complies with the constraints defined in Equations (4)–(10) as it is a feasible solution. In particular, the reliability Equation (10) satisfies the expression in Equation (17), completing the proof.

$$R_K\left(G_S\left(\left\{x_{i,j}\right\}\right)\right)\ge p_{min}\iff R_V\left(\mathcal{T}\right)\ge p_{min}\iff \prod _{e\in \mathcal{T}}{p}_e\ge p_{min}$$

(17)

□

7. Conclusions and Future Work

This article introduced the Generalized Steiner Problem with Node-Connectivity Constraints under the hostile reliability model. This is a relevant symmetric network design problem with application in nowadays technological and scientific communities. A hybrid metaheuristic approach combining GRASP and VND was proposes to efficient solve the addressed problem.

A reliability analysis was performed under the hostile model, where both nodes and links are subject to probabilistic failures. The analysis studied the relationship between the network optimization and the reliability evaluation. The research was based on relevant questions related to the number of feasible networks found by the proposed GRASP/VND algorithm for the full probabilistic model, a sensitivity analysis of the proposed model with respect to elementary probabilities of operation for both edges and nodes, and a sensitivity analysis of the proposed model with respect to the connectivity constraints defined for terminal nodes.

The main results demonstrated the efficacy of the proposed VND search, which achieved relative cost improvements above 35% over the solution built in the construction phase. Regarding the proposed research questions, the main answers based on the experimental results reported in Section 5 are as follows. RQ 1: A very high number of feasible solutions were computed by GRASP/VND, even reaching 100% of computed solutions meting the reliability threshold in all but three problem instances solved, for the considered input parameters $(p_{min},P_E,P_{V-T})$. RQ 2: Two different situations were detected when varying the elementary reliabilities of links and Steiner nodes: fixing the elementary reliability for nodes and reducing the elementary reliabilities for links resulted in a significant reduction of the number of feasible solutions. Fixing the link reliabilities and varying the node reliabilities, the reduction of the resulting network reliability was less than in the previous case. RQ 3: Results demonstrated a proportional increase of the number of networks that meet the reliability threshold with respect to the connectivity constraints defined for terminal nodes. Networks with a high number of imposed disjoint paths have also a highly robust topology, i.e., their reliability levels are high. RQ 4: Overall, for the hostile model, the empirical analysis indicate that it is better improving the elementary probabilities of operation of links than improving the elementary probabilities of Steiner nodes, to meet a required reliability threshold for the designed network.

The obtained results can be applied to any type of network with associated design/construction costs where the elementary probabilities of operation of the links and the optional (Steiner) nodes are known. Some relevant examples include the design of electrical power networks and fiber optic networks in telecommunications.

Finally, two particular cases of the GSPNCHR that can be solved in polynomial time were presented. Theoretical results were developed to compute optimal solutions in polynomial times and polynomial algorithms to solve them were presented.

The main lines for future work are related to extend the experimental evaluation of the proposed hybrid metaheuristic approach and design improved search operators to address larger problem instances. The sensitivity analysis of perturbations can be extended too, including scenarios that consider the theoretical results for problem instances solvable in polynomial time.