Content uploaded by Jin-Kao Hao
Author content
All content in this area was uploaded by Jin-Kao Hao on Nov 28, 2022
Content may be subject to copyright.
A Fast Tri-individual Memetic Search Approach for the
Distance-based Critical Node Problem
Yangming Zhoua,b, Gezi Wangc, Jin-Kao Haod, Na Genga,b,
Zhibin Jianga,b,∗
aData-Driven Management Decision Making Lab, Shanghai Jiao Tong University,
Shanghai 200030, China
bSino-US Global Logistics Institute, Antai College of Economics and Management,
Shanghai Jiao Tong University, Shanghai 200030, China
cDepartment of Computer Science and Engineering, East China University of Science
and Technology, Shanghai 200237, China
dDepartment of Computer Science, Universit´e d’Angers, Angers 49045, France
European Journal of Operational Research, Nov. 2022
https://doi.org/10.1016/j.ejor.2022.11.039
Abstract
The distance-based critical node problem involves identifying a subset
of nodes in a graph such that the removal of these nodes leads to a resid-
ual graph with the minimum distance-based connectivity. Due to its NP-
hard nature, solving this problem using exact algorithms has proved to be
highly challenging. Moreover, existing heuristic algorithms are typically
time-consuming. In this work, we introduce a fast tri-individual memetic
search approach to solve the problem. The proposed approach maintains a
small population of only three individuals during the whole search. At each
generation, it sequentially executes an inherit-repair recombination opera-
tor to generate a promising offspring solution, a fast betweenness centrality-
based late-acceptance search to find high-quality local optima, and a simple
population updating strategy to maintain a healthy population. Extensive
experiments on both real-world and synthetic benchmarks show our method
significantly outperforms state-of-the-art algorithms. In particular, it can
steadily find the known optimal solutions for all 22 real-world instances
with known optima in only one minute, and new upper bounds on the re-
∗Corresponding author
Email addresses: yangming.zhou@sjtu.edu.cn (Yangming Zhou),
y80200226@mail.ecust.edu.cn (Gezi Wang), jin-kao.hao@univ.angers.fr (Jin-Kao
Hao), gengna@sjtu.edu.cn (Na Geng), zbjiang@sjtu.edu.cn (Zhibin Jiang)
Preprint submitted to Elsevier November 28, 2022
maining 22 large real-world instances. For 54 synthetic instances, it finds
new upper bounds on 36 instances, and matches the previous best-known
upper bounds on 15 other instances in ten minutes. Finally, we investigate
the usefulness of each key algorithmic ingredient.
Keywords: Combinatorial optimization; Critical node problem;
Distance-based connectivity; Metaheuristic; Memetic search.
1. Introduction
Graphs or networks arise in a variety of application areas due to their
elegance and inherent ability to logically describe important interactive rela-
tions. In a number of situations, they are greatly affected by a small fraction
of influential nodes whose removal would significantly degrade certain net-
work functionality. To identify these important nodes (referred to as critical
nodes), critical node detection problems (CNDPs) [9,39,20,47,50,11]
have been proposed and widely studied across a variety of domains. Take
epidemic control as an example, the decision-maker is often interested in
identifying a limited number of people to be vaccinated to reduce the overall
transmissibility of disease virus [14,19]. To monitor carbon dioxide emis-
sions, the decision-maker attempts to find the critical paths and nodes that
contribute strongly to carbon emissions embodied in transmission [44,45].
The critical node problem (CNP) is a well-known representative CNDP
[9]. It aims to minimize the pairwise connectivity of the residual graph via
the removal of a limited subset of nodes from the original graph. The pair-
wise connectivity measure essentially counts the number of pairs of nodes
connected by a path. As indicated in [32], the existence of a path be-
tween two nodes may not be sufficient, and the path length should also
be considered as well (as short as possible). Indeed, considering the path
length is especially relevant to model practical applications in supply chain
networks, communication networks, and transportation networks, where a
distance-based connectivity measure needs to be optimized. Correspond-
ingly, distance-based critical node problems have been studied in recent
years [42,7,22,32].
The distance-based critical node problem (DCNP) considered in this
work is a computationally challenging NP-hard problem [42]. Compared to
CNP, few efforts have been devoted to developing efficient algorithms for
DCNP in the literature. In addition, existing heuristic algorithms are time-
consuming [4,32]. To enrich the solution approaches of DCNP, we propose
2
a fast tri-individual memetic search (FTMS) approach. Our contributions
are two fold.
The proposed fast tri-individual memetic search approach is charac-
terized by the use of a small population of only three individuals. At
each generation, it sequentially executes an inherit-repair recombina-
tion operator (to generate an offspring solution), a fast betweenness
centrality-based late-acceptance search procedure (to perform local op-
timization), and a simple population updating strategy (to manage the
population).
We conduct extensive experiments on both real-world and synthetic
benchmarks to evaluate FTMS. Comparative results show that FTMS
competes well with state-of-the-art algorithms in terms of both solu-
tion quality and computation time. In particular, it can steadily and
quickly find the optimal solutions for all 22 real-world instances with
known optima in only one minute, and new upper bounds for 22 large
real-world instances. It significantly outperforms state-of-the-art al-
gorithms on 54 synthetic instances, finding new upper bounds on 36
instances and matching previous best-known upper bounds on 15 other
instances in ten minutes.
The rest of this paper is organized as follows. After an introduction
of DCNP in Section 2, we conduct a brief overview of previous studies on
both CNP and DCNP in Section 3. Section 4presents the proposed FTMS
method for DCNP. Sections 5and 6are devoted to performance comparisons
and experimental analyses, respectively. Finally, we conclude in Section 7.
2. Problem Description
Given an undirected graph G= (V, E) with n=|V|nodes (vertices) and
m=|E|edges, and a positive integer B(i.e., budgetary constraint), DCNP
aims to remove a subset of nodes of cardinality at most Bto minimize a
distance-based connectivity metric, i.e.,
min
S⊆VX
i,j∈V\S,i<j
cij ·ψ(d(i, j)) (1)
s.t. X
i∈S
wi≤B(2)
where d(i, j) is the distance (i.e., the length of the shortest path) between
nodes iand jin the residual graph G[V\S], ψ:Z+∪ {+∞} → Rrepresents
3
a distance-based connectivity metric, cij ∈R+is the cost associated with
the connection between the pair of nodes iand j,wi∈R+is the weight of
node i∈V. For any two disconnected nodes iand jin G[V\S], we have
d(i, j)=+∞.
The distance-based connectivity measure ψ(·) of a graph is assumed to
be a function of the actual pairwise distances between nodes in the remain-
ing graph (e.g., global efficiency, Harary index, characteristic path length,
Wiener index) [42] instead of simply knowing whether nodes are connected
or not in the classic CNP. Several DCNPs were studied in the literature
based on different distance-based connectivity measures, e.g., minimizing
the total number of pairs of nodes connected by a hop distance of at most
k; minimizing the Harary index, or equivalently, the efficiency of the graph;
minimizing the sum of power functions of distances in the graph; maximiz-
ing the generalized Wiener index, and maximizing the shortest path between
two given nodes in the graph.
In this work, we consider the general case of DCNP, i.e., minimizing
the number of node pairs connected by a path of length at most k. Cor-
respondingly, the distance-based pairwise connectivity metric is defined as
follows:
ψ(d(i, j)) = (1, d(i, j )≤k,
0,otherwise,(3)
where d(i, j) is the distance (i.e., the length of the shortest path) between
nodes iand jin the residual graph G[V\S], and kis a given positive
integer indicating a maximal distance limit. It means that only when the hop
distance between a pair of nodes does not exceeds k, they can be considered
as a truly connected node pair. Here, we focus on the unweighted version
of DCNP, i.e., cij = 1, ∀i, j ∈Vand i < j. Correspondingly, the objective
function of DCNP can be further described as follows:
f(S) = X
i,j∈V\S,i<j
ψ(d(i, j)) S⊆V(4)
When k= 1, DCNP reduces to minimizing the number of remaining
edges, which is also known as the maximum coverage problem (MCP) [10].
The goal of MCP is to find a subset S⊆Vwith a fixed number of nodes
and the number of edges covered by Sis maximized, which is equivalent to
minimize the number of remaining edges. When k≥n−1, DCNP reduces
to minimizing the total number of connected node pairs, which is the classic
CNP [9,48]. Both MCP and CNP are known to be NP-hard [9,10].
4
3. Related Work
3.1. Previous Studies on CNP
Detecting critical nodes in complex network is a challenging combina-
torial optimization problem. Due to its NP-hard nature, it has attracted
much attention and considerable efforts have been dedicated to address this
important problem [9,16,35,38,1,47]. Existing algorithms can be divided
into two categories: exact and heuristic algorithms.
Exact algorithms are known to guarantee the optimality of their obtained
solution. Arulselvan et al. [9] presented the first integer programming (IP)
model for CNP and solved it with the CPLEX solver. Di Summa et al.
[16] further proposed two improved IP models, and solved them within the
framework of the branch-and-cut method. They could find optimal solu-
tions for small sparse instances with up to 150 nodes. Veremyev et al. [41]
developed an improved compact IP model for CNP, which was able to op-
timally solve CNP instances with up to 1200 nodes. They further proposed
a general IP model for CNP and its variants [40], which could provide opti-
mal solutions only for medium instances up to an instance with 1612 nodes
and 2106 edges in affordable computation time. Ventresca and Aleman [36]
proposed a randomized rounding algorithm for the cardinality-constrained
CNP, which achieved a 1/(1 −θ)-approximation. These exact or approxi-
mation algorithms are practical only on instances of limited sizes, typically
with no more 5000 nodes.
To handle large instances, heuristic algorithms are good alternatives to
provide approximate solutions in a reasonable computation time. Two cat-
egories of heuristic algorithms have been proposed in the literature. The
first category is single solution-based methods, which manipulate only a
single candidate solution, such as constructive heuristics [29,37,1], simu-
lated annealing [34], variable neighborhood search [6,33] and iterated local
search [6,43]. Another category is population-based methods, which main-
tain a population of candidate solutions that are manipulated and evolved
during the search process, such as population-based incremental learning
[34], genetic algorithm [5], path-relinking [30], memetic algorithm [48] and
its variant named variable population memetic algorithm [47]. Compared
to single solution-based methods, population-based methods show better
performance for CNP. To our knowledge, most of the best-known results
available in the literature were achieved by memetic algorithms. However,
they suffer from the need of managing a large population to maintain the
diversity of the search, making them typically time-consuming, in particular
5
when they are applied to large and very large instances with at least several
thousands of nodes.
3.2. Previous Studies on DCNP
Like CNP, DCNP is also a computationally challenging NP-hard problem
[42]. Compared to CNPs, much less efforts have been made on DCNP.
Existing algorithms for DCNP include exact and heuristic algorithms.
To solve optimally DCNP, Verremyev et al. [42] proposed two exact al-
gorithms. The first one relies on an integer programming formulation (IP)
with additional preprocessing and modeling enhancements. However, for
some types of distance-based metrics, the model size grows quickly with
the number of nodes, thus the standard solver like CPLEX can only handle
very small instances. For solving larger instance, the same authors developed
a truncate-and-resolve algorithm (TRA) that iteratively solved a series of
simplified IPs to obtain an optimal solution of the original problem. Exper-
imental results showed TRA significantly outperforms (by at least an order
of magnitude) the algorithm based on the initial IP formulation. TRA can
optimally solve instances with about 500 nodes. With the help of variable
fixing rules, sparse instances with 1500 nodes are also solved. However,
TRA has two main drawbacks. On the one hand, it is sensitive to the di-
ameter of the graph, and it would be time-consuming for instances with
large diameter. On the other hand, it is sensitive to the length of the edges,
and it performs badly on instances with the large length of the edges. To
overcome the above issues, Hooshmand et al. [22] proposed a new IP model
with simple structure and solved it by an efficient Bender decomposition
algorithm, which is better than TRA in terms of computational time. This
algorithm is not sensitive to the graph diameter nor length of the edges.
Recently, Salemi and Buchanan [32] introduced two new IP models (i.e.,
thin and path-like). Comparative results between their models and the IP
model of [42] showed that these three models were equivalent in strength
when the objective coefficients were non-negative, but the thin model was
the strongest generally. Although the thin model generally has an exponen-
tial number of constraints, it admits an efficient separation routine used in
a branch-and-cut algorithm. Alozie et al. [3] presented a new path-based
model (PBM) for DCNP, where separation heuristics and valid inequalities
that exploit the structure of the problems were used to enhance the model.
Besides general graphs, Aringhieri et al. [8] presented dynamic program-
ming algorithms for DCNP over special graph classes such as paths, trees,
and series-parallel graphs.
6
To solve DCNP approximately, Aringhieri et al. [7] performed a pre-
liminary analysis and provided some suggestions on designing heuristic al-
gorithms. Alozie et al. [4] presented a centrality-based heuristic (CBH)
algorithm for DCNP, which combined a backbone-based crossover proce-
dure to generate an offspring solution and a centrality-based neighborhood
search to improve it. However, it is computationally expensive, which makes
it unpractical for solving hard and large DCNP instances. To enrich the
heuristic solution approaches for DCNP, we are devoted to proposing an
efficient heuristic algorithm to solve DCNP.
4. Fast Tri-individual Memetic Search for DCNP
This section presents the fast tri-individual memetic search (FTMS) ap-
proach for DCNP. It starts with the solution representation and evaluation,
followed by the overall framework and a detailed introduction of each algo-
rithmic module.
4.1. Solution Representation and Evaluation
Given a graph G= (V, E) with n=|V|nodes and m=|E|edges, and
an integer budget B, any subset S⊆Vwith no more than Bnodes is a
feasible candidate solution of DCNP, i.e., |S| ≤ B. Let S(|S|< B) be
a candidate solution, it is easy to verify that any solution S0←S∪ {v}
with one more node v∈V\Sis a feasible solution no worse than S, i.e.,
f(S0)≤f(S). Since Bis the largest integer for a solution to be feasible,
we can safely consider only candidate solutions Swith exactly Bnodes.
Therefore, we represent a solution of DCNP as S={vS(1), vS(2), . . . , vS(B)}
where S(i) (1 ≤i≤B) is the index of i-th node in S. Therefore, the search
space Ω contains all possible subsets S⊆Vsuch that |S|=B.
Given a candidate solution S, its objective function value f(S) can be
directly computed according to equation (4), which counts the total number
of node pairs connected by a hop distance equal to or less than kin the
residual graph G[V\S]. This evaluation requires O(n3) time even by the
fastest algorithm. To reduce its complexity, an alternative method is applied
based on the k-depth breadth first search (BFS) tree built for each node
in O(mn). As indicated in [4], this time complexity can be significantly
improved to O(bk) when the depth of BFS tree is limited to k, where b
denotes the branching factor of the tree.
7
4.2. Overall Framework
FTMS is a hybrid evolutionary algorithm and follows the general memetic
algorithm (MA) approach [28] that combines local search and population-
based search. MAs have been successfully applied to solve many NP-hard
problems, such as the graph coloring problem [26], the orienteering problem
with hotel selection [18], the critical node problem [48], the maximum di-
versity problem [46], and the soft-clustered vehicle routing problem [49] and
the multiple traveling repairman problem with profits [31]. The canonical
memetic algorithms often maintain a relatively large population (at least 10
individuals) during the search. It is very time-consuming to manage such a
large population. Recently, some effort have been made to develop efficient
memetic algorithms based on small population.
Inspired by [27,17], FTMS maintains a small population of only three in-
dividuals, whose rationale is explained in Section 4.3. The overall framework
of FTMS is shown in Algorithm 1. FTMS consists of four main modules:
population initialization, inherit-repair recombination (IRR), betweenness
centrality-based late-acceptance search (BCLS) and population updating.
Once the small initial population is built by the population initialization
procedure, the algorithm enters a loop to form a number of generations.
At each generation, a promising offspring solution is first generated by the
IRR operator, and it is then improved to a high-quality local optimum by
the fast BCLS procedure. Then the improved offspring solution is used to
update the population. The process repeats until a given stopping condi-
tion is satisfied, e.g., the computation time reaches a given time limit or the
number of generations exceeds an allowable maximal generation count.
4.3. Population Initialization
FTMS starts its search from a small initial population of only three
distinct individuals, i.e., P={S1, S2, S3}. As centrality metrics evaluate
the importance of a node for various definitions of importance [7], S1,S2
and S3are constructed based on three different centrality metrics:
Degree centrality (χD) is a simple count of the total number of edges
incident to a target node. It cannot recognize a difference between
quantity and quality.
Katz centrality (χK) [23] computes the centrality for a target node
based on the centrality of its neighbors. It is a generalization of the
eigenvector centrality.
8
Algorithm 1: Pseudo-code of FTMS
Input: A DCNP instance with the budgetary constraint B,
maximal idle iteration count ˆ
ζ, maximal idle generation
count ˆ
ξand selection probability φ
Output: The best solution S∗found
1// build an initial population
2P={S1, S2, S3} ← PopulationInitialization()
3S∗←arg min∀Si∈Pf(Si)
4ξ←0 /*idle generation count*/
5while Stopping condition is not met do
6// generate an offspring solution by an IRR operator
7S0←IRR(P)
8// improve it by a local search
9S←BCLS(S0,ˆ
ζ, φ)
10 if f(S)< f(S∗)then
11 S∗←S
12 // update the population
13 P←PopulationUpdating(P, S, ξ, ˆ
ξ)
14 return The best solution S∗found
Betweenness centrality (χB) [12] is a measure of centrality in a graph
based on shortest paths. It measures the fraction of shortest paths
passing through a target node. The target node would have a high
betweenness centrality if it appears in many shortest paths. As in-
dicated in [12], all betweenness centrality values can be computed in
O(mn) under hop-based distances and in O(mn+n2log n) under edge-
weighted distances.
These three centralities are popular measures, and they have been widely
used to evaluate the importance of a node [24]. Degree centrality is the
simplest centrality measure to compute, which can be quickly obtained in
O(n). For both Katz and betweenness centralities, we consider their special
cases: k-Katz centrality and k-betweenness centrality. The former ranks
nodes based on the size of the k-depth BFS tree rooted at each node, while
the latter ranks nodes according to the number of their direct offspring
summed over all generated k-depth BFS trees. Once the BFS trees have
been built, k-Katz and k-betweenness centralities can be computed in O(n)
and O(mn), respectively [13].
9
5
1
3 42
(a) An undirected graph
1 2 4
3
5
(b) A tree rooted at node 3
1 2
4
1
5
3 4
32 4
(c) Short paths
1
Figure 1: An Illustrative Example of Degree Centrality, k-Katz Centrality and k-
Betweenness Centrality)
To well understand the above-mentioned centrality metrics, we illustrate
them with an example shown in Figure 1. Figure 1(a) presents an undirected
graph Gof five nodes, Figure 1(b) is a tree rooted at node 3, and Figure. 1(c)
lists four shortest paths whose length does not exceed kand endpoint does
not contain node 3. Given node 3 and the maximal hop distance k= 2,
its degree centrality value equals the number of neighboring nodes, i.e.,
χD= 3. The Katz centrality value of node 3 counts the number of nodes
whose distance to node 3 does not exceed k, which is equivalent to reducing
the size of the tree rooted at node 3 by one, i.e., χK= 4. The betweenness
centrality value of node 3 is the sum of the ratios of the number of shortest
paths (marked as blue) passing through node 3 and corresponding distance
does not exceed kto the total number of shortest paths whose distance does
not exceed k, i.e., χB= 2.
From an empty initial solution S1, we first sort all nodes in Gaccording
to their degree centrality χDvalues in descending order, and then each
node is iteratively added to S1with the probability δ(0.5≤δ < 1) until
S1contains Bnodes, i.e., |S1|=B. Correspondingly, S2and S3can be
constructed based on the k-Katz centrality and k-betweenness centrality,
respectively.
4.4. Inherit-repair Recombination
At each generation, FTMS employs an inherit-repair recombination (IRR)
operator to construct a promising offspring solution. IRR shares similar
ideas with backbone-based crossovers used in [48,4]. It works in two stages:
random inheritance and greedy repair stages.
10
Definition 1. (First Backbone). First backbone is a set of common nodes
shared by all three solutions (i.e., S1,S2and S3), which can be formally
defined as Rst =S1∩S2∩S3.
Definition 2. (Second Backbone). Second backbone consists of common
nodes shared by only two solutions, which is formally defined as Rnd =
((S1∩S2)∪(S1∩S3)∪(S2∩S3)) \ Rst.
Definition 3. (Third Backbone). Third backbone is composed of nodes that
belong to only one solution, which can be formally defined as Rrd = (S1∪
S2∪S3)\(Rst ∪ Rnd).
Given the three distinct solutions S1,S2and S3in P, their nodes can be
divided into three subsets denoted by the first backbone, second backbone
and third backbone, as shown in Definitions 1-3. According to the defini-
tions, three kinds of backbones can be identified from S1,S2and S3, and
they form the set of all nodes in P, i.e., S1∪S2∪S3=Rst ∪ Rnd ∪ Rrd . In
addition, we define the set of nodes that does not belong to any solution of
Pas the non-backbone, i.e., Rno =V\(S1∪S2∪S3).
At the random inheritance stage, a partial solution S0is first obtained
by directly inheriting all nodes of Rst, i.e., S0← Rst . If the size of S0is
less than floor(η·B), at each step, a node is randomly added to S0from a
chosen backbone until |S0| ≥ floor(η·B), where 0 < η < 1 is a proportional
factor. At each step, a backbone is selected based on two pre-defined factors
θ(0.5≤θ < 1) and ϕ(0.5≤ϕ < 1). Specifically, the second backbone
Rnd is selected with the probability θ, and the probabilities to select the
third backbone Rrd and non backbone Rno are (1 −θ)ϕand (1 −θ)(1 −ϕ),
respectively.
At the greedy repair stage, the partial offspring solution S0is repaired by
greedily adding a node from the remaining nodes (i.e., V\S0) until a feasible
offspring solution is obtained. In particular, a node uis selected if it leads
to the best improvement to f(S), i.e., u←arg max{f(S0)−f(S0∪ {v})},
∀v∈V\S0.
The IRR operator treats the common nodes shared by parent solutions
as good elements, and aims to transmit the common nodes into the par-
tial offspring solution at the random inherit stage. To ensure the quality of
the offspring solution, the greedy repair stage of the IRR operator greedily
repairs the partial solution to a feasible solution. The IRR operator can
also be considered as an improved backbone-based crossover of [4]. It dis-
tinguishes itself from the backbone-based crossover by the node selection
11
strategy. In particular, the backbone-based crossover adopts a hybrid node
selection strategy by randomly or greedily selecting a node with a proba-
bility, while IRR employs a random and greedy node selection strategy in
the inheritance and repair phases, respectively. Given the way that IRR re-
combines two parent solutions to obtain an offspring solution, this operator
ensures simultaneously the role of search diversification and intensification
of the FTMS algorithm.
4.5. Betweenness Centrality-based Late-acceptance Search
FTMS integrates a fast local optimization procedure called betweenness
centrality-based late-acceptance search (BCLS) (see Algorithm 2). At the
beginning, a node sequence Lis constructed based on the k-betweenness
centrality of nodes in the residual graph G[V\S], which takes time O(mn +
nlog n). Specifically, all nodes in G[V\S] are first sorted according to
their k-betweenness centrality values in descending order. Then, the top-λ
nodes are stored in the linked list L, where λ=B+ max{5,floor(0.2∗B)}
like [4]. The remaining nodes are added into Lin a random way. Once
the node sequence Lis obtained, BCLS enters a loop to iteratively perform
node exchanges between Sand L(lines 5-24). Each exchange operation is
realized in two steps with the add and remove operators. The former aims
to add a node uof Linto S, while the latter tries to greedily remove a node
vfrom S. For each head node uof L, we add it into Swith a pre-defined
selection probability φ(0.5≤φ < 1). Once node uis added into S, we
greedily remove a node vfrom Sin O(Bbk), which minimally deteriorates
the objective function f(S). With probability 1 −φ, node uis re-inserted
into an intermediate position (i.e., L.begin() + 5) of L(line 24). The whole
computational complexity of BCLS is O(mn +nlog n+˜
ζBbk), where ˜
ζis
the number of iterations (i.e., iteration count).
BCLS distinguishes itself from the centrality-based neighborhood search
(CNS) [4] in two aspects. On the one hand, BCLS always selects a node to
insert into Saccording to the k-betweenness centrality only. On the other
hand, BCLS adopts a linked list data structure to represent a node sequence
(i.e., L) obtained based on the k-betweenness centrality, which delays the
acceptance of a node. These features ease the neighborhood operations
during the search.
4.6. Population Updating
To maintain the diversity of the three individual population, a simple
pool updating strategy is employed in FTMS. Once an improved offspring
12
Algorithm 2: Pseudo-code of BCLS
Input: A solution S, maximal idle iteration count ˆ
ζand selection
probability φ
Output: A improved solution S∗
1S∗←S
2// construct a node sequence L
3Initialize Lbased on the k-betweenness centrality
4ζ←0 /*idle iteration count*/
5while ζ < ˆ
ζdo
6// record the head node of L
7u← L.front
8// remove the head node from L
9L.pop(u)
10 Generate a random decimal r∈[0,1]
11 if r < φ then
12 S←S∪ {u}
13 v←arg minw∈S{f(S\ {w})−f(S)}
14 S←S\ {v}
15 // add node vto the end of L
16 L.push(v)
17 if f(S)< f(S∗)then
18 S∗←S
19 ζ←0
20 else
21 ζ←ζ+ 1
22 else
23 // insert node uinto a new position of L
24 L.insert(L.begin() + 5, u)
25 return An improved solution S∗
solution is obtained, it is accepted or discarded according to the population
updating strategy, shown in Algorithm 3.
Given an improved offspring solution Sand the current population P,
if Sis the same as an individual of P, we discard it. Otherwise, Sreplaces
the worst individual Swof Punder two conditions, i.e., either Sis better
than the worst individual Swin Por the idle update count ξreaches the
13
Algorithm 3: Pseudo-code of Population Updating
Input: Population P, an improved offspring S, idle update count ξ
and maximal idle update count ˆ
ξ
Output: An updated population P
1if Sdoes not exist in Pthen
2// identify the worst individual
3Sw←arg max∀Si∈Pf(Si)
4// replace the worst one
5if f(S)< f(Sw)or ξ > ˆ
ξthen
6P←P∪ {S}\{Sw}
7ξ←0
8else
9ξ←ξ+ 1
10 else
11 ξ←ξ+ 1
12 return An updated population P
allowable maximal idle update count ˆ
ξ.
4.7. Computational Complexity of FTMS
To analyze the computational complexity of FTMS, we consider four
main modules of Algorithm 1. FTMS starts the search from a small popu-
lation Pof three distinct solutions generated by the population initialization
procedure of the time complexity of O(mn +nlog n), where n=|V|and
m=|E|present the total number of nodes and edges in G, respectively.
For each subsequent generation, FTMS sequentially executes the IRR
operator, BCLS procedure, and population updating strategy. An offspring
solution can be obtained in O(Bη +bk+B(1 −η)nbk) by the IRR operator,
where 0 < η < 1 is a proportional factor to control the size of the partial so-
lution at the random inheritance stage of IRR, kand bdenote the depth and
branching factor of the BFS tree, respectively. Then, the BCLS procedure
is applied to improve Sby performing neighborhood search around it, which
can be finished in O(mn +nlog n+˜
ζBbk), where ˜
ζis the iteration count of
BCLS. Once an improved offspring solution is found, the population is up-
dated with it in O(B) time. Therefore, the total time complexity of FTMS
is O(Bη +bk+B(1 −η)nbk+mn +nlog n+˜
ζBbk) for each generation.
14
5. Computational Experiments
5.1. Benchmark Instances
Our computational experiments were conducted on benchmark instances
used in recent studies [42,3,4]. In addition, 11 large real-world CNP in-
stances [5] are first adapted for DCNP. These instances consists of two cat-
egories: real-world and synthetic benchmarks.
Real-world benchmark are divided into two categories: R1 and R2.
The former is composed of 11 real-world networks selected from the
Pajek and UCINET1datasets. The latter consists of 11 large instances
selected from CNP instances2.
Synthetic benchmark are further classified into two groups: S1 and
S2. The former contains 21 instances (i.e., Barabasi-Albert, Erdos-
Renyi, and uniform random graphs) generated by using NetworkX
random graph generators [21], while the latter includes two original
CNP instances (i.e., FF250 and WS250a) and 10 new instances. These
new instances share the same sizes and orders as the original bench-
mark instances in [34].
The main characteristics of both real-world and synthetic instances are
presented in Table 1. Following [4], we solve each instance of the synthetic
benchmark S2 with a given Bvalue, while for the remaining instances, we
solve each instance under two different budgetary constraints, i.e., B=
floor(0.05n) and B= floor(0.1n).
5.2. Experimental Settings
Our algorithms3are implemented in the C++ programming language
and complied with gcc 8.1.0 and the flag “-O0”. All experiments are carried
out on a computer equipped with an AMD Ryzen 7 5800U processor with 1.9
GHz and 16 GB RAM operating under the Windows 10 system. In following
experiments, we set the largest hop distance as three, i.e., k= 3. It is an
appropriate hop distance for most of benchmark instances, as suggested in
[42,34].
1http://vlado.fmf.uni-lj.si/pub/networks/data/
2http://individual.utoronto.ca/mventresca/cnd.html
3Our programs and results are available at https://github.com/YangmingZhou/DCNPs
15
Table 1: Characteristics of Benchmark Instances
Real-world Instances Synthetic Instances
Name n m Density(%) Name n m Density(%)
Hi tech 33 91 0.172 ba1 100 475 0.096
Karate 34 78 0.139 ba2 100 900 0.182
Mexican 35 117 0.197 er1(3) 80 474 0.150
Sawmill 36 62 0.098 er1(6) 80 476 0.151
Chesapeake 39 170 0.229 er1(9) 80 447 0.141
Dolphins 62 159 0.084 er2(3) 200 982 0.049
Lesmiserable 77 254 0.087 er2(6) 200 1018 0.051
Santafe 118 200 0.029 er2(9) 200 1017 0.051
Sanjuansur 75 155 0.056 gnm1 200 1000 0.050
LindenStrasse 232 303 0.011 gnm2 300 1500 0.033
USAir97 332 2126 0.039 gnm3 300 2000 0.045
yeast 2018 2705 0.001 FF250 250 514 0.017
Ham1000 1000 1998 0.004 BA250 250 1225 0.039
Ham2000 2000 3996 0.002 BA500 500 2475 0.020
Ham3000a 3000 5999 0.001 BA1000 1000 4975 0.010
Ham3000b 3000 5997 0.001 ER250 250 1190 0.038
Ham3000c 3000 5996 0.001 ER500 500 2570 0.021
Ham3000d 3000 5993 0.001 ER1000 1000 999 0.002
Ham3000e 3000 5996 0.001 WS250a 250 1246 0.040
Ham4000 4000 7997 0.001 WS250b 250 1250 0.040
Ham5000 5000 6594 0.001 WS500 500 2500 0.020
powergrid 4941 6594 0.001 GNM250 500 1250 0.010
GNM500 1000 2500 0.005
5.3. Parameter Tuning
Our experimental results are obtained by executing FTMS algorithm
with the parameter settings provided in Table 2. To determine the suit-
able parameter values, we resort to the well-known automatic parameter
configuration tool called IRACE [25].
Table 2: Parameter Settings of FTMS
Parameter Description Candidate Values Final Value Section
δProbability to Add a Node to an Initial Solution {0.5,0.6,0.7,0.8,0.9}0.8 4.3
ηProportion Factor {0.1,0.3,0.5,0.7,0.9}0.9 4.4
θFirst Factor to Select a Backbone {0.5,0.6,0.7,0.8,0.9}0.5 4.4
ϕSecond Factor to Select a Backbone {0.5,0.6,0.7,0.8,0.9}0.9 4.4
ˆ
ζMaximal Idle Iteration Count {50,100,150,200,250}150 4.5
φProbability to Add Node uto S{0.5,0.6,0.7,0.8,0.9}0.8 4.5
ˆ
ξMaximal Idle Generation Count {3,5,7,9,11}54.6
For each parameter, it requires some candidate values as input, as shown
in the column “Candidate Values” of Table 2. The best parameter configura-
tion is provided in the column of “Final Value”. During the parameter tun-
ing, we run IRACE with the default settings, and set the total time budget
as 2000 executions. The whole experiments are conducted on eight represen-
tative instances with different sizes that are selected from both real-world
and synthetic benchmarks, i.e., USAir97, ba2(6), er1(3), er2(9), gnm2(3),
16
gnm3(9), ER250, and WS500. For each instance, it is solved with the time
limit ˆ
t= 60 seconds.
5.4. Compared with State-of-the-art Algorithms
To evaluate the performance of FTMS, we experimentally compare it
with three state-of-the-art (SOTA) algorithms, i.e., the exact algorithm
called path-based model (PBM) [3], the centrality-based heuristic (CBH)
algorithm [4] and its multi-start version, i.e., multi-start centrality-based
heuristic (MCBH). Since the source codes of CBH and MCBH are not avail-
able to us, we have re-implemented them in C++. To guarantee a fair
comparison, we execute each algorithm on the same computational plat-
form with the following stopping conditions. For each benchmark instance,
we run each algorithm ten times with the time limit. Since there is no an
available time limit in the literature, we determine the time limit based on
the preliminary results. Specifically, we execute CBH with the time limit
ˆ
t= 3600 seconds at each run. For each instance of real-world benchmark R1
and synthetic benchmark S1, we run both MCBH and FTMS under the time
limit ˆ
t= 60 seconds for each run. For each instance of real-world benchmark
R2 and synthetic benchmark S2, we adopt the time limit ˆ
t= 600 seconds.
Note that the results of PBM are directly obtained from [3], which were
achieved under the time limit ˆ
t= 3600 seconds.
To analyze the comparative results, we resort to the well-known Wilcoxon
signed rank test [15] to check the significant difference on each comparison
indicator between two compared algorithms. At a significance level of 0.05,
algorithm Xis significantly better than algorithm Yif its p-value is no more
than 0.05.
5.4.1. Results on Real-world Benchmark Instances
Comparative results between FTMS and the state-of-the-art algorithms
on real-world benchmark R1 with B= floor(0.05n) and B= floor(0.10n) are
summarized in Table 3and Table 4, respectively. In these tables, column
1 presents the instance name (Instance), column 2 indicates the optimal
solutions (f∗) of PBM, reported in [3]. Columns 3-5 show the results of
CBH, including the best result ( ˆ
f) found during 10 runs, the average re-
sult ( ¯
f) and average computation time (¯
t) in seconds needed to reach the
best result at each run. Correspondingly, columns 6-8 and 9-11 present the
results of MCBH and FTMS, respectively. In addition, we count the num-
ber of instances in which FTMS finds better (#Wins), equal (#Ties) and
worse (#Loses) results compared to each reference algorithm. The last row
provides the p-values of the Wilcoxon signed ranks test.
17
Table 3: Comparison of FTMS and SOTA Algorithms on Real-world Benchmark R1 with
B= floor(0.05n)
PBM CBH MCBH?FTMS
Instance f∗ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
Hi tech 397∗397 397.0 0.2 397 397.0 0.1 397 397.0 0.1
Karate 324∗324 324.0 0.2 324 324.0 0.1 324 324.0 0.1
Mexican 527∗527 527.0 0.3 527 527.0 0.1 527 527.0 0.1
Sawmill 215∗215 215.0 0.2 215 215.0 0.1 215 215.0 0.1
Chesapeake 696∗696 696.0 0.3 696 696.0 0.1 696 696.0 0.1
Dolphins 820∗820 820.0 1.3 820 820.0 0.2 820 820.0 0.1
Lesmiserable 930∗930 930.0 1.9 930 930.0 0.1 930 930.0 0.1
Santafe 305∗305 305.0 1.2 305 305.0 0.1 305 305.0 0.1
Sanjuansur 803∗803 803.0 0.9 803 803.0 0.1 803 803.0 0.1
LindenStrasse 1054∗1054 1057.8 5.9 1054 1054.0 1.2 1054 1054.0 0.1
USAir97 10623∗10623 10697.2 269.7 10623 10623.0 23.2 10623 10623.0 5.4
#Wins|Ties|Loses 0|11|0 0|11|0 2|9|0 11|0|0 0|11|0 0|11|0 3|8|0 – – –
#p-value 1.0e0 1.0e0 5.0e-1 – 1.0e0 1.0e0 – – – –
∗Optimal results obtained by the exact algorithm reported in [3] within 3600 seconds.
?MCBH denotes a multi-start version of CBH.
From Table 3, we observe that all 11 real-world instances can be opti-
mally solved by almost all three heuristic algorithms, i.e., CBH, MCBH and
FTMS. In particular, FTMS can steadily find the optimal solutions for all
11 instances in the shortest time. For USAir97 instance, FTMS obtains the
optimal solution in 5.4 seconds, against 269.7 and 23.2 seconds for CBH and
MCBH, respectively. Compared to MCBH, FTMS finds better results on
3 instances and the same results on the remaining instances in terms of ¯
t.
FTMS achieves better results than CBH on all 11 instances in terms of ¯
t.
At a significance level of 0.05, there is no significant performance difference
among CBH, MCBH and FTMS in terms of both ˆ
fand ¯
fon real-world
benchmark R1.
We can obtain similar observations from Table 4. FTMS is able to
steadily find the optimal solutions on all real-world instances except US-
Air97. For USAir97 instance, all three heuristic algorithms can find the
optimal solution, but FTMS also obtains the smallest ¯
fvalue. Although
all 11 real-world instances can be optimally solved by each heuristic algo-
rithm, FTMS can reach them in the shortest computation time with the
highest success rate. At a significance level of 0.05, FTMS significantly out-
performs CBH in terms of both ¯
fand ¯
t, and it is also significantly better
than MCBH in terms of ¯
t. Results from both Tables 3and 4show FTMS
competes favorably with SOTA algorithms in terms of both solution quality
and computation time.
To further evaluate the performance of FTMS, we experimentally com-
pare FTMS with SOTA on real-world benchmark R2. Detailed comparative
18
Table 4: Comparison of FTMS and SOTA Algorithms on Real-world Benchmark R1 with
B= floor(0.1n)
PBM CBH MCBH?FTMS
Instance f∗ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
Hi tech 293∗293 294.8 0.5 293 293.0 0.9 293 293.0 0.1
Karate 147∗147 150.9 0.4 147 147.0 0.6 147 147.0 0.1
Mexican 358∗358 358.0 0.6 358 358.0 0.1 358 358.0 0.1
Sawmill 135∗135 135.0 0.2 135 135.0 0.1 135 135.0 0.1
Chesapeake 512∗512 515.2 1.1 512 512.0 0.1 512 512.0 0.1
Dolphins 583∗583 591.7 2.5 583 583.0 1.3 583 583.0 0.1
Lesmiserable 323∗323 323.0 2.9 323 323.0 0.1 323 323.0 0.1
Santafe 116∗116 116.0 2.8 116 116.0 0.1 116 116.0 0.1
Sanjuansur 457∗457 457.2 2.0 457 457.0 1.2 457 457.0 0.1
LindenStrasse 429∗429 431.7 12.7 429 429.0 5.2 429 429.0 0.1
USAir97 3100∗3100 3219.1 423.6 3100 3180.1 32.2 3100 3110.1 6.7
#Wins|Ties|Loses 0|11|0 0|11|0 7|4|0 11|0|0 0|11|0 1|10|0 6|5|0 – – –
#p-value 1.0e0 1.0e0 1.6e-2 – 1.0e0 1.0e0 – – – –
∗Optimal results obtained by exact algorithm reported in [3] within 3600 seconds.
?MCBH denotes a multi-start version of CBH.
results between FTMS and SOTA algorithms on real-world benchmark R2
with B= floor(0.05n) and B= floor(0.10n) are summarized in Tables 5and
6, respectively. From them, we observe that FTMS also performs very well
on real-world benchmark R2. In particular, FTMS finds the best results
in terms of ˆ
ffor all instances of R2. In terms of ¯
f, FTMS finds the best
results for all instances except for Hamilton1000. At a significance level of
0.05, FTMS significantly outperforms CBH and MCBH in terms of both ˆ
f
and ¯
f.
5.4.2. Results on Synthetic Benchmark Instances
To further evaluate the performance of FTMS, we show computational
results on the two sets of synthetic instances. Comparative results between
FTMS and SOTA algorithms on synthetic benchmark S1 are summarized
in Tables 7-8with the same information as in the previous section.
As we can see from Table 7, FTMS reaches an excellent performance. It
finds new upper bounds for 12 instances, and matches the previous upper
bounds for 6 out of the remaining 9 instances. At a significance level of
0.05, we observe that FTMS significantly outperforms CBH in terms of all
performance indicators (i.e. ˆ
f,¯
fand ¯
f). FTMS is also significantly better
than MCBH in terms of both ¯
fand ¯
t. While for ˆ
f, there is no significant
difference between FTMS and MCBH.
From Table 8, we observe that FTMS also demonstrates a good perfor-
mance. It finds new upper bounds on 15 instances, and matches the previous
19
Table 5: Comparison of FTMS and SOTA Algorithms on Real-world Benchmark R2 with
B= floor(0.05n)
CBH MCBH?FTMS
Instance ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
yeast1 4231 4262.4 44.5 4222 4263.0 44.2 4197 4197.0 35.3
Hamilton1000 18185 18206.0 9.8 18181 18189.4 281.8 18181 18204.0 13.3
Hamilton2000 37086 37151.9 77.7 37050 37138.2 81.3 37021 37040.3 102.5
Hamilton3000a 55449 55499.9 266.3 55443 55506.6 268.5 55402 55413.3 234.7
Hamilton3000b 55354 55392.6 255.7 55379 55416.7 234.5 55318 55343.4 223.6
Hamilton3000c 55566 55614.5 254.0 55571 55624.6 331.2 55514 55544.5 359.3
Hamilton3000d 55477 55513.2 254.0 55447 55634.5 453.4 55404 55425.7 243.5
Hamilton3000e 56177 56220.7 259.7 56145 56275.2 346.1 56104 56156.7 282.7
Hamilton4000 74956 75075.9 572.5 74931 75020.1 550.1 74893 74938.1 489.2
Hamilton5000 98538 98819.5 0.8 98427 98792.3 0.8 93253 93324.8 442.6
powergrid 20584 21141.8 409.9 20563 20623.2 363.3 20533 20546.6 248.9
#Wins|Ties|Loses 11|0|0 11|0|0 – 10|1|0 10|0|1 – – – –
#p-value 9.8e-4 9.8e-4 –2.0e-3 2.0e-3 – – – –
?MCBH denotes a multi-start version of CBH.
Table 6: Comparison of FTMS and SOTA Algorithms on Real-world Benchmark R2 with
B= floor(0.1n)
CBH MCBH?FTMS
Instance ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
yeast1 1397 1421.1 56.1 1397 1421.1 51.9 1380 1382.1 75.5
Hamilton1000 11873 11917.0 19.3 11884 11929.5 20.6 11850 11862.9 36.7
Hamilton2000 24532 24604.9 147.3 24526 24619.4 147.6 24430 24475.4 264.4
Hamilton3000a 36710 40990.2 60.6 36304 36393.3 466.0 36175 36223.9 568.4
Hamilton3000b 36187 36301.2 523.6 36231 36872.0 452.7 36081 36128.9 559.5
Hamilton3000c 36591 36670.2 494.6 36537 38453.1 331.7 36425 36503.7 548.1
Hamilton3000d 36525 36588.2 521.5 36478 37544.8 456.5 36362 36430.9 565.8
Hamilton3000e 36786 36907.6 500.8 36822 36943.5 486.6 36697 36748.9 564.3
Hamilton4000 56742 57301.2 0.5 56890 57350.8 0.5 49772 49876.2 567.3
Hamilton5000 71467 71875.9 0.8 71083 71697.0 0.7 62366 62595.2 597.2
powergrid 10836 10943.3 598.5 10709 10828.3 593.0 10667 10681.3 531.0
#Wins|Ties|Loses 11|0|0 11|0|0 – 11|0|0 11|0|0 – – – –
#p-value 9.8e-4 9.8e-4 –9.8e-4 9.8e-4 – – – –
?MCBH denotes a multi-start version of CBH.
20
Table 7: Comparison of FTMS and SOTA Algorithms on Synthetic Benchmark S1 with
B= floor(0.05n)
PBM CBH MCBH?FTMS
Instance LB UB ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
ba1(3) 4275∗4275 4275 4275.0 8.5 4275 4275.0 0.7 4275 4275.0 0.1
ba1(6) 4278∗4278 4278 4278.4 9.8 4278 4278.0 0.2 4278 4278.0 0.1
ba1(9) 4193∗4193 4193 4193.0 7.7 4193 4193.0 0.2 4193 4193.0 0.1
ba2(3) 4384 4461 4465 4465.0 20.9 4465 4465.0 0.1 4465 4465.0 0.1
ba2(6) 4369∗4369 4436 4453.4 19.3 4371 4409.4 19.4 4371 4371.0 1.6
ba2(9) 4371 4463 4465 4465.0 18.3 4465 4465.0 0.1 4465 4465.0 0.1
er1(3) 2798 2835 2842 2843.4 9.6 2835 2838.5 25.0 2835 2835.0 3.9
er1(6) 2799 2835 2835 2839.6 11.2 2835 2835.0 4.0 2835 2835.0 0.5
er1(9) 2814∗2814 2847 2847.7 8.4 2814 2819.9 16.7 2814 2815.6 6.8
er2(3) 15990 16955 16842 16897.3 101.5 16823 16837.4 27.4 16818 16829.2 13.4
er2(6) 16026 16930 16887 16930.0 91.5 16833 16838.6 25.4 16829 16831.4 12.6
er2(9) 15970 16954 16899 16900.9 129.1 16761 16787.3 20.9 16761 16764.9 19.1
gnm1(3) 15972 16771 16706 16716.6 136.4 16638 16649.5 27.6 16638 16657.2 6.7
gnm1(6) 16209 17062 16975 16977.4 104.6 16965 16965.2 27.6 16965 16967.4 19.0
gnm1(9) 16099 16958 16843 16860.1 89.0 16843 16843.0 9.8 16843 16844.5 7.7
gnm2(3) 34015 36803 35332 35334.7 281.4 35332 35346.8 32.7 35332 35337.4 17.6
gnm2(6) 33701 36445 35236 35245.4 303.4 35191 35219.1 30.8 35203 35215.9 30.2
gnm2(9) 33782 36641 35331 35350.8 353.9 35298 35309.2 26.7 35303 35303.0 17.2
gnm3(3) 36403 40229 39978 39982.0 627.3 39620 39702.0 39.2 39555 39615.6 44.5
gnm3(6) 36557 40217 39848 39876.1 681.6 39490 39546.2 34.9 39334 39440.0 52.1
gnm3(9) 36258 40176 39852 39880.5 605.1 39548 39597.5 38.8 39544 39578.7 59.0
#Wins|Ties|Loses – 12|6|3 13|8|0 16|4|1 – 5|14|2 12|6|3 – – – –
#p-value – 2.2e-3 1.5e-3 4.2e-4 – 3.5e-1 5.0e-2 – – – –
∗Optimal results obtained by exact algorithm reported in [3] within 3600 seconds.
?MCBH denotes a multi-start version of CBH.
21
Table 8: Comparison of FTMS and SOTA Algorithms on Synthetic Benchmark S1 with
B= floor(0.1n)
PBM CBH MCBH?FTMS
Instance LB UB ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
ba1(3) 3330∗3330 3330 3330.0 13.6 3330 3330.0 0.8 3330 3330.0 0.2
ba1(6) 3390∗3390 3390 3391.0 14.1 3390 3390.0 0.7 3390 3390.0 0.2
ba1(9) 3328∗3328 3328 3328.4 13.6 3328 3328.0 2.3 3328 3328.0 0.4
ba2(3) 3716 3987 4005 4005.0 46.1 4002 4004.3 11.0 3916 3991.2 20.9
ba2(6) 3718 3916 3955 3955.9 44.5 3916 3918.8 18.0 3909 3910.5 20.8
ba2(9) 3702 3986 4004 4004.0 48.6 4002 4004.2 9.9 3916 3949.7 18.8
er1(3) 2395 2474 2535 2538.7 20.5 2474 2474.6 17.8 2474 2475.8 1.0
er1(6) 2395 2482 2485 2519.1 22.9 2482 2482.3 19.2 2482 2482.1 1.4
er1(9) 2378 2452 2528 2539.0 20.6 2452 2456.5 24.1 2452 2457.5 2.8
er2(3) 12469 14886 14332 14360.0 213.5 14326 14341.7 31.3 14331 14344.9 21.7
er2(6) 12575 15052 14364 14380.2 217.2 14262 14329.6 39.6 14225 14235.7 14.6
er2(9) 12414 15038 14397 14434.0 255.7 14333 14368.3 30.2 14347 14363.5 22.4
gnm1(3) 12517 14730 14193 14256.3 252.2 14161 14173.1 34.4 14161 14177.2 17.2
gnm1(6) 12601 14658 14399 14419.0 221.4 14393 14420.5 40.0 14393 14399.4 15.1
gnm1(9) 12565 14803 14195 14211.3 178.1 14184 14186.3 26.1 14184 14184.4 13.5
gnm2(3) 25877 28978 28715 28734.4 637.4 28715 28723.9 32.8 28715 28715.0 22.7
gnm2(6) 25262 30635 28554 28629.7 647.9 28541 28584.3 32.9 28540 28540.4 38.9
gnm2(9) 25765 30805 28868 28922.1 709.9 28872 28912.0 37.0 28823 28833.3 52.5
gnm3(3) 28541 35847 35158 35198.5 1539.6 34903 35014.2 37.9 34733 34886.5 56.6
gnm3(6) 27307 35501 35000 35079.6 1353.8 34635 34762.4 41.8 34552 34596.6 57.0
gnm3(9) 28698 35704 34785 34837.7 1323.1 34505 34725.1 40.1 34522 34641.0 60.0
#Wins|Ties|Loses – 15|6|0 17|4|0 20|1|0 – 8|10|3 14|3|4 – – – –
#p-value – 6.5e-4 2.9e-4 9.0e-5 –5.0e-2 2.5e-3 – – – –
∗Optimal results obtained by exact algorithm reported in [3] within 3600 seconds.
?MCBH denotes a multi-start version of CBH.
22
best-known upper bounds on the remaining six instances. At a significance
level of 0.05, FTMS significantly outperforms the state-of-the-art algorithms
in terms of all performance indicators (i.e., ˆ
f,¯
fand ¯
t).
Finally, comparative results between FTMS and SOTA algorithms on
synthetic benchmark S2 are summarized in Tables 9. It is worth noting that
each instance is solved with a fixed Bvalue like in [4]. Both MCBH and
FTMS are executed with a longer time limit, i.e., ˆ
t= 600 seconds.
Table 9: Comparison of FTMS and SOTA Algorithms on Synthetic Benchmark S2
PBM CBH MCBH?FTMS
Instance BLB UB ˆ
f¯
f¯
tˆ
f¯
f¯
tˆ
f¯
f¯
t
FF250 13 1587∗1587 1587 1598.2 16.7 1587 1587.0 0.9 1587 1587.0 0.1
BA250 25 13772∗13772 13772 13788.3 137.6 13772 13773.2 24.5 13772 13772.0 7.5
BA500 50 24847∗24847 24847 24847.0 1104.7 25316 31325.3 18.1 24847 24847.0 25.2
BA1000 100 16071 316735 59178 60488.9 3600.0 58651 72576.7 475.4 58493 58528.2 234.9
ER250 25 17959 22288 19894 19931.6 326.5 19870 19870.0 133.2 19870 19872.4 58.2
ER500 50 30846 79482 68062 68129.2 3189.6 68012 68098.9 222.3 68012 68020.8 145.5
ER1000 100 70494 221831 173538 174326.1 3600.0 175773 177637 0.3 170351 170671.5 380.7
WS250a 70 1039 2319 2034 2056.8 728.7 1889 1919.4 255.0 1882 1919.5 27.9
WS250b 25 14586 15223 15020 15044.7 316.1 15020 15022.6 384.8 15020 15029.3 97.4
WS500 50 25907 53729 51460 51567.2 3483.4 51500 51539.3 246.3 51422 51434.7 126.1
GNM250 25 18638 22711 20967 20984.5 453.8 20944 20944.0 151.7 20944 20944.0 64.8
GNM500 50 29462 78001 65775 65892.9 2883.0 65857 65925.0 185.5 65775 65816.4 206.3
#Wins|Ties|Loses – – 9|3|0 7|5|0 11|1|0 – 6|6|0 7|2|3 – – – –
#p-value – – 2.0e-4 4.4e-4 8.9e-5 –6.5e-4 4.6e-4 – – – –
∗Optimal results obtained by exact algorithm reported in [3] within 3600 seconds.
?MCBH denotes a multi-start version of CBH.
From Table 9, we find that FTMS also show an excellent performance
on synthetic benchmark S2. Specifically, it attains new upper bounds for 9
instances, and matches the previous best-known upper bounds for the re-
maining three instances. At a significance level of 0.05, it is significantly
better than the state-of-the-art algorithms in terms of all performance indi-
cators, i.e., ˆ
f,¯
fand ¯
t. These results show FTMS competes well with SOTA
algorithms.
5.5. Results of FTMS with a Long Time Limit ˆ
t= 600 Seconds
To further study the behavior of FTMS algorithm, we report the results
of FTMS with a long time limit ˆ
t= 600 seconds. As observed from Tables
3-7, FTMS is able to steadily find the optimal solutions for all instances of
the real-world benchmark R1 in only 60 seconds. It also can reach the known
optimal results of the first three instances of the synthetic benchmark S1
both with B= floor(0.05n) and B= floor(0.1n). Therefore, our experiment
focuses on the remaining instances that have not been optimally solved by
FTMS with a 100% success rate.
Comparative results between FTMS with ˆ
t= 60 and ˆ
t= 600 seconds are
summarized in Tables 10-11. In these tables, column 1 gives the instance
23
name (Instance), columns 2-3 present the lower bounds (LB) and upper
bounds (UB), respectively. Columns 4-6 provide the results of FTMS under
ˆ
t= 60, including ˆ
f,¯
fand ¯
t. Correspondingly, columns 7-9 show the results
of FTMS under ˆ
t= 600 seconds. In addition, we calculate the number
of instances on which FTMS finds a better (#Wins), equal (#Ties) and
worse (#Loses) results in terms of both ˆ
fand ¯
fcompared to each reference
algorithm. The last row provides the p-values of the Wilcoxon signed ranks
test.
Table 10: Results of FTMS on Synthetic Benchmark S1 with B= floor(0.05n)
PBM FTMS (ˆ
t= 60s) FTMS (ˆ
t= 600s)
Instance LB UB ˆ
f¯
f¯
tˆ
f¯
f¯
t
ba2(3) 4384 4461 4465 4465.0 0.0 4465 4465.0 0.0
ba2(6) 4369 4369 4371 4371.0 1.6 4371 4371.0 2.0
ba2(9) 4371 4463 4465 4465.0 0.0 4465 4465.0 0.0
er1(3) 2798 2835 2835 2835.0 3.9 2835 2835.0 4.8
er1(6) 2799 2835 2835 2835.0 0.5 2835 2835.0 0.8
er1(9) 2814 2814 2814 2815.6 6.8 2814 2814.0 12.7
er2(3) 15990 16955 16818 16829.2 13.4 16818 16818.0 15.3
er2(6) 16026 16930 16829 16831.4 12.6 16829 16834.0 22.8
er2(9) 15970 16954 16761 16764.9 19.1 16761 16761.0 37.8
gnm1(3) 15972 16771 16638 16657.2 6.7 16638 16638.0 8.1
gnm1(6) 16209 17062 16965 16967.4 19.0 16965 16967.1 28.2
gnm1(9) 16099 16958 16843 16844.5 7.7 16843 16844.5 11.8
gnm2(3) 34015 36803 35332 35337.4 17.6 35332 35337.4 115.1
gnm2(6) 33701 36445 35203 35215.9 30.2 35191 35198.2 115.6
gnm2(9) 33782 36641 35303 35303.0 17.2 35298 35302.0 38.5
gnm3(3) 36403 40229 39555 39615.6 44.5 39473 39555.9 84.5
gnm3(6) 36557 40217 39334 39440.0 52.1 39315 39350.9 149.4
gnm3(9) 36258 40176 39544 39578.7 59.0 39371 39460.5 94.0
#Wins|Ties|Loses – 12|3|3 5|13|0 10|7|1 – – – –
#p-value – 1.3e-3 2.7e-2 3.9e-2 – – – –
From Table 10, we observe that FTMS improves its results under a
longer time limit ˆ
t= 600 seconds. In particular, it finds new upper bounds
for 12 instances, and matches the previous upper bounds for 3 out of the
remaining 6 instances. Compared to FTMS with ˆ
t= 60 seconds, FTMS with
ˆ
t= 600 seconds demonstrates a better performance in terms of both ˆ
fand
¯
f. At a significance level of 0.05, FTMS with ˆ
t= 600 seconds significantly
outperforms FTMS with ˆ
t= 60 seconds in terms of both ˆ
fand ¯
f. Moreover,
we find FTMS quickly converges to local optimum in no more than 150
seconds for all instances.
As we can see from Table 11, FTMS also improves its performance on
synthetic benchmark S1 with B= floor(0.1n). It finds new upper bounds
for 15 instances and matches the previous upper bounds on the remaining
three instances. At a significance level of 0.05, FTMS with ˆ
t= 600 seconds
24
Table 11: Results of FTMS on Synthetic Benchmark S1 with B= floor(0.1n)
PBM FTMS (ˆ
t= 60s) FTMS (ˆ
t= 600s)
Instance LB UB ˆ
f¯
f¯
tˆ
f¯
f¯
t
ba2(3) 3716 3987 3916 3991.2 20.9 3916 3944.4 76.6
ba2(6) 3718 3916 3909 3910.5 20.8 3909 3909.0 52.5
ba2(9) 3702 3986 3916 3949.7 18.8 3916 3916.0 68.3
er1(3) 2395 2474 2474 2475.8 1.0 2474 2474.0 5.3
er1(6) 2395 2482 2482 2482.1 1.4 2482 2482.2 1.1
er1(9) 2378 2452 2452 2457.5 2.8 2452 2453.7 3.9
er2(3) 12469 14886 14331 14344.9 21.7 14320 14333.6 59.9
er2(6) 12575 15052 14225 14235.7 14.6 14225 14225.0 21.0
er2(9) 12414 15038 14347 14363.5 22.4 14344 14347.5 54.9
gnm1(3) 12517 14730 14161 14177.2 17.2 14161 14162.2 27.8
gnm1(6) 12601 14658 14393 14399.4 15.1 14393 14394.5 57.7
gnm1(9) 12565 14803 14184 14184.4 13.5 14184 14184.0 16.6
gnm2(3) 25877 28978 28715 28715.0 22.7 28715 28715.0 49.5
gnm2(6) 25262 30635 28540 28540.4 38.9 28540 28540.8 83.6
gnm2(9) 25765 30805 28823 28833.3 52.5 28823 28831.0 160.3
gnm3(3) 28541 35847 34670 34813.7 76.5 34581 34676.2 243.3
gnm3(6) 27307 35501 34552 34596.6 57.0 34418 34449.8 177.1
gnm3(9) 28698 35704 34522 34641.0 60.2 34466 34498.0 193.8
#Wins|Ties|Loses – 15|3|0 5|13|0 15|1|2 – – – –
#p-value – 4.3e-4 2.7e-2 3.2e-4 – – – –
is also significantly better than FTMS with ˆ
t= 60 seconds in terms of both
ˆ
fand ¯
f. These results confirm that FTMS is able to find still better results
under a long time limit. Furthermore, we find FTMS converges to local
optimum in no more than 250 seconds for all instances.
6. Experimental Analysis
In this section, we perform additional experiments to gain a deeper un-
derstanding of FTMS. In particular, we perform four groups of experiments:
1) to compare the run-time distributions of FTMS and the state-of-the-art
algorithm MCBH, 2) to evaluate the benefit of the IRR operator, 3) to
investigate the effectiveness of the BCLS procedure, and 4) to analyze the
randomness of FTMS. Our experimental analyses are conducted on the eight
representative instances used for parameter tuning.
6.1. Run-time Distributions of FTMS and MSCH
To further evaluate FTMS, we resort to the time-to-target (TTT) plots
[2] to analyze the run-time distributions of both FTMS and MCBH on the
eight representative instances. A TTT plot is a useful tool for algorithm
performance comparison. To produce a TTT plot, we run each algorithm
100 times, and record the computation time to obtain a solution at least
25
as good as a given target value. After sorting them in ascending order, a
probability piis associated with the i-th computation time ti. A TTT plot is
obtained by plotting these points (ti, pi), i = 1,2,...,100. Figure 2presents
the TTT plots of FTMS and MCBH, where the target value of each instance
is given in the parentheses behinds its instance name.
0 20 40
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
USAir97 (3350)
FTMS
MCBH
0 20 40 60
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
ba2(6) (3920)
FTMS
MCBH
0246
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
er1(3) (2845)
FTMS
MCBH
0 20 40 60
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
er2(9) (16800)
FTMS
MCBH
0 10 20 30
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
gnm2(3) (35370)
FTMS
MCBH
0 20 40 60
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
gnm3(9) (34915)
FTMS
MCBH
0 20 40 60
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
ER250 (19910)
FTMS
MCBH
0 20 40 60 80
Time to target value (s)
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative probability
WS500 (52518)
FTMS
MCBH
Figure 2: Run-time Distributions of FTMS and MCBH Algorithms
From Figure 2, we can observe that FTMS is more likely to find a target
solution faster than MCBH. For USAir97 instance, it shows that the prob-
ability of finding the target value 3350 at most 20 seconds is approximately
60% for MCBH, while it is 100% for FTMS. Taking the gnm2(3) instance as
another example, we find that the probabilities of finding the target value
35370 at most 10 seconds are about 20% and 90% for MCBH and FTMS,
respectively. Similar observations apply to the other six instances. These
results clearly show that FTMS outperforms MCBH.
6.2. Superiority of the IRR Operator
FTMS uses the inherit-repair recombination operator to generate off-
spring solutions. To show its interest, we experimentally compare FTMS
with a variant called FTMS0. FTMS0is obtained from FTMS by replac-
ing the IRR operator with the backbone-based crossover presented in [4].
26
Comparative results between FTMS and FTMS0are summarized in Table
12. At its bottom, we calculate the number of instances for which FTMS
finds better (#Wins), equal (#Ties) and worse (#Loses) results compared
to FTMS0.
Table 12: Comparison between FTMS0(with backbone-based crossover) and FTMS (with
inherit-repair recombination operator)
FTMS0FTMS
Instance Bˆ
f¯
f¯
tˆ
f¯
f¯
t
USAir97 33 3100 3100.0 20.5 3100 3100.0 11.4
ba2(6) 10 3909 3910.3 37.9 3909 3909.7 27.5
er1(3) 4 2835 2835.0 1.9 2835 2835.0 3.5
er2(9) 10 16761 16778.5 12.3 16761 16764.9 19.1
gnm2(3) 15 35332 35339.5 24.4 35332 35338.3 17.5
gnm3(9) 30 34559 34713.6 45.9 34475 34621.9 53.3
ER250 25 19870 19872.4 32.7 19870 19872.4 23.2
WS500 50 51532 51663.9 56.3 51483 51639.0 35.4
#Wins|Ties|Loses – 2|6|0 5|3|0 7|0|1 – – –
From Table 12, we observe that FTMS performs better than FTMS0.
FTMS finds better results on two instances, and the same results on the
remaining six instances in terms of ˆ
f. In terms of ¯
f, FTMS obtains better
results on five instances and the same results on the remaining three in-
stances. For the computation time, FTMS also remains competitive. These
results confirm the interest of the inherit-repair recombination operator.
6.3. Effectiveness of the BCLS Procedure
To evaluate the effectiveness of the betweenness centrality-based late-
acceptance search (BCLS), we compare FTMS with an alternative algorithm
named FTMS00. FTMS00 is obtained from FTMS by replacing BCLS with the
centrality-based neighborhood search (CNS) proposed in [4]. Comparative
performances in terms of the best result and average result are presented
in the left and right sides of Figure 3, respectively. The x-axis indicates
the instance name, and y-axis displays the performance gaps. By treating
FTMS00 as a baseline algorithm, we calculate their performance gap as (f−
˜
f)/˜
f×100%, where fis the result of FTMS and ˜
fis the result of FTMS00.
A performance gap smaller than zero means that FTMS achieves a better
result on the corresponding instance.
As we can see from the left side of Figure 3, FTMS has a better perfor-
mance than FTMS00 in terms of the best result. In particular, it finds better
27
USAir97
ba2(6)
er1(3)
er2(9)
gnm2(3)
gnm3(9)
ER250
WS500
6
5
4
3
2
1
0
Gap to the result of FTMS"(×10 3)
best result
FTMS"(with CNS)
FTMS(with BCLS)
USAir97
ba2(6)
er1(3)
er2(9)
gnm2(3)
gnm3(9)
ER250
WS500
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Gap to the result of FTMS"(×10 3)
average result
FTMS"(with CNS)
FTMS(with BCLS)
Figure 3: Comparison between FTMS00 (with centrality-based neighborhood search) and
FTMS (with betweenness centrality-based late-acceptance search)
results on three instances and the same results on the remaining five in-
stances. FTMS also demonstrates an excellent performance in terms of the
average result, as shown in the right side of Figure 3. We can observe that
FTMS is able to achieve better results on six instances and the same result
on the remaining two instances. These results conform the effectiveness of
BCLS.
6.4. Randomness Analysis of FTMS
Randomization is common in many implementations of metaheuristics.
Our FTMS algorithm also integrates several randomized components, such
as the IRR operator and the BCLS procedure. Taking BCLS as an exam-
ple, we experimentally evaluate the interest of randomization. As shown in
Algorithm 2, at each iteration, a head node uof Lis added into Swith a
probability φ. Otherwise, node uis re-inserted into an intermediate position
of L. To show the merit of this randomized strategy in FTMS, we exper-
imentally compare FTMS with an alternative version named FTMS000 by
setting φ= 1.0. That is, a head node uis always added into Sat each iter-
ation of BCLS. Figure 4describes the comparative performance of FTMS000
and FTMS.
From Figure 4, we observe that FTMS performs better in terms of the
best result on three instances than FTMS000, and they have the same per-
formance on the remaining five instances. In terms of the average result,
28
USAir97
ba2(6)
er1(3)
er2(9)
gnm2(3)
gnm3(9)
ER250
WS500
6
5
4
3
2
1
0
Gap to the result of FTMS'''(×10 3)
best result
FTMS'''(less randomness)
FTMS
USAir97
ba2(6)
er1(3)
er2(9)
gnm2(3)
gnm3(9)
ER250
WS500
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Gap to the result of FTMS'''(×10 2)
average result
FTMS'''(less randomness)
FTMS
Figure 4: Comparison between FTMS000 (with φ= 1.0) and FTMS (with φ= 0.8)
FTMS also shows a better performance by finding improved results on six
instances and the same results on the remaining two instances as FTMS000
does. These observations confirm the usefulness of randomization in FTMS.
7. Conclusion and Future Work
Detecting critical nodes in a complex network represents a class of chal-
lenging NP-hard problems. Considerable efforts have been devoted to de-
veloping efficient algorithms for critical node detection problems. However,
few efforts have been made to solve the distance-based critical node prob-
lems, which aim to identify a subset of nodes in a network whose deletion
minimizes the distance-based pairwise connectivity (i.e., the number of node
pairs connected by a path of length at most k). To address it, we proposed a
fast tri-individual memetic search (FTMS) method. FTMS is characterized
by a small population of only three individuals, which relies on an inherit-
repair recombination operator to generate a promising solution and a fast
betweenness centrality-based late-acceptance search to find high-quality lo-
cal optima during the search.
Extensive computational studies on both real-world and synthetic bench-
marks show FTMS is highly competitive compared to state-of-the-art algo-
rithms. In particular, FTMS is able to steadily find the optimal solutions for
22 real-world instances with known optima in only one minute, and new up-
per bounds for the remaining 22 large real-world instances. For 54 synthetic
29
instances, FTMS also achieves excellent performance by finding 36 new up-
per bounds and matching 15 previous upper bounds. As future work, it
is worthy of further improving FTMS based on other centrality measures,
such as closeness centrality and eigenvector centrality. It is also interest-
ing to adopt FTMS for solving other distance-based critical node detection
problems.
Acknowledgment
We would like to thank the anonymous referees for their helpful com-
ments and suggestions, which helped us to improve the presentation of the
work. This work was supported in part by the National Natural Science
Foundation of China under Grants 61903144, 71871142 and 71931007, and
Startup Plan for New Young Teachers of Shanghai Jiao Tong University
under Grant 22X010503820.
Appendix A. Influence of the Population Size
Unlike the general memetic algorithm [28], the FTMS algorithm main-
tains a small population of only three individuals. To investigate the in-
fluence of the population size on the performance of FTMS, we experimen-
tally compare the three-individual FTMS with FTMS using 10 individuals
(named FTMS1). Detailed comparative results are summarized in Table
A.13.
Table A.13: Comparison of FTMS Algorithms with Different Population Sizes
FTMS1(with large population) FTMS (with small population)
Instance Bˆ
f¯
f¯
tˆ
f¯
f¯
t
USAir97 33 3100 3109.9 23.2 3100 3100.0 11.4
ba2(6) 10 3909 3910.9 28.7 3909 3909.7 27.5
er1(3) 4 2835 2835.0 2.4 2835 2835.0 3.5
er2(9) 10 16761 16765.6 30.8 16761 16764.9 19.1
gnm2(3) 15 35353 35353.0 12.9 35332 35338.3 17.5
gnm3(9) 30 34534 34662.4 55.9 34475 34621.9 53.3
ER250 25 19870 19881.3 35.7 19870 19872.4 23.2
WS500 50 51588 51782.0 33.5 51483 51639.0 35.4
#Wins|Ties|Loses – 3|5|0 7|1|0 7|0|1 – – –
The results in Table A.13 show that FTMS performs better than FTMS1
in terms of both the best and average results. In particular, FTMS obtains
better results than FTMS1in terms of ˆ
f, including improved results on three
instances and same results on the remaining five instances. In terms of ¯
f,
30
FTMS also has a better performance by finding improved results on all in-
stances except er1(3) than FTMS1, For er1(3), both FTMS and FTMS1find
the same result. These results confirm the usefulness of the small population
strategy used in FTMS.
Appendix B. Rationale behind the Population Initialization Strat-
egy
The FTMS algorithm starts its search from a small population consisting
of three distinct individuals, i.e., S1,S2and S3. Those three individuals
are constructed based on the degree centrality, k-Katz centrality and k-
betweenness centrality, respectively. To show the usefulness behind this
population initialization strategy, we experimentally compare FTMS with
six variants using different population initialization strategies as follows.
FTMS2: the initial population is built only based on the degree cen-
trality;
FTMS3: the initial population is built only based on the k-Katz cen-
trality;
FTMS4: the initial population is built only based on the k-betweenness
centrality;
FTMS5: the initial population is built based on both the degree and
k-Katz centralities;
FTMS6: the initial population is built based on both the degree and
k-betweenness centralities;
FTMS7: the initial population is built based on both the k-Katz and
k-betweenness centralities.
Table B.14 summarizes the comparative results between FTMS and its
six variants. At its bottom, we give the average value and average rank of
both ˆ
fand ¯
fperformance indicators. We order these seven algorithms for
each instance separately, the best performing algorithm obtaining the rank
of 1, the second best rank of 2, and so on. In case of ties, average ranks are
assigned. Finally, we obtain the average rank of each algorithm by averaging
the ranks of all eight instances. The smaller the average rank, the better the
algorithm. From the results, we observe that FTMS obtains the smallest
average values and the smallest average ranks in terms of both ˆ
fand ¯
f. In
31
Table B.14: Comparison of FTMS Algorithms with Different Initialization Strategies
FTMS2FTMS3FTMS4FTMS5FTMS6FTMS7FTMS
Instance Bˆ
f¯
fˆ
f¯
fˆ
f¯
fˆ
f¯
fˆ
f¯
fˆ
f¯
fˆ
f¯
f
USAir97 33 3100 3215.5 3100 3186.5 3100 3174.0 3100 3163.9 3100 3228.0 3100 3163.9 3100 3100.0
ba2(6) 10 3909 3910.0 3909 3910.0 3909 3910.2 3909 3910.7 3909 3909.2 3909 3910.1 3909 3909.7
er1(3) 4 2835 2835.0 2835 2835.0 2835 2835.0 2835 2835.0 2835 2835.0 2835 2835.0 2835 2835.0
er2(9) 10 16761 16763.7 16761 16770.7 16761 16766.7 16761 16762.8 16761 16768.6 16761 16767.6 16761 16764.9
gnm2(3) 15 35341 35351.8 35332 35343.1 35341 35348.2 35332 35347.3 35341 35349.4 35332 35344.9 35332 35338.3
gnm3(9) 30 34497 34668.0 34553 34683.5 34542 34619.4 34606 34665.8 34613 34648.0 34562 34636.0 34475 34621.9
ER250 25 19870 19874.1 19870 19876.4 19870 19871.0 19870 19875.2 19870 19874.8 19870 19874.8 19870 19872.4
WS500 50 51566 51791.0 51492 51632.6 51466 51594.0 51564 51816.2 51627 51760.4 51473 51565.7 51483 51639.0
avg.value 20984.9 21051.1 20981.5 21029.7 20978.0 21014.8 20997.1 21047.1 21007.0 21046.7 20980.3 21012.3 20970.6 21010.2
avg.rank 4.3 4.7 3.8 4.8 3.8 3.4 4.2 4.6 5.0 4.7 3.7 3.5 3.3 2.4
particular, FTMS finds the best solutions on all test instances except for
WS500. For the WS500 instance, FTMS obtains the third best solution.
These results prove the usefulness of our chosen population initialization
strategy.
References
References
[1] Addis, B., Aringhieri, R., Grosso, A., Hosteins, P., 2016. Hybrid con-
structive heuristics for the critical node problem. Annals of Operations
Research 238, 637–649.
[2] Aiex, R.M., Resende, M.G., Ribeiro, C.C., 2007. TTT plots: a perl
program to create time-to-target plots. Optimization Letters 1, 355–
366.
[3] Alozie, G.U., Arulselvan, A., Akartunalı, K., Pasiliao Jr, E.L., 2021.
Efficient methods for the distance-based critical node detection problem
in complex networks. Computers & Operations Research 131, 105254.
[4] Alozie, G.U., Arulselvan, A., Akartunalı, K., Pasiliao Jr, E.L., 2022. A
heuristic approach for the distance-based critical node detection prob-
lem in complex networks. Journal of the Operational Research Society
73, 1347–1361.
[5] Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R., 2016a. A
general evolutionary framework for different classes of critical node
problems. Engineering Applications of Artificial Intelligence 55, 128–
145.
32
[6] Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R., 2016b. Local
search metaheuristics for the critical node problem. Networks 67, 209–
221.
[7] Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R., 2016c. A
preliminary analysis of the distance based critical node problem. Elec-
tronic Notes in Discrete Mathematics 55, 25–28.
[8] Aringhieri, R., Grosso, A., Hosteins, P., Scatamacchia, R., 2019. Poly-
nomial and pseudo-polynomial time algorithms for different classes of
the distance critical node problem. Discrete Applied Mathematics 253,
103–121.
[9] Arulselvan, A., Commander, C.W., Elefteriadou, L., Pardalos, P.M.,
2009. Detecting critical nodes in sparse graphs. Computers & Opera-
tions Research 36, 2193–2200.
[10] Ausiello, G., Boria, N., Giannakos, A., Lucarelli, G., Paschos, V.T.,
2012. Online maximum k-coverage. Discrete Applied Mathematics 160,
1901–1913.
[11] Baggio, A., Carvalho, M., Lodi, A., Tramontani, A., 2021. Multilevel
approaches for the critical node problem. Operations Research 69, 486–
508.
[12] Brandes, U., 2001. A faster algorithm for betweenness centrality. Jour-
nal of Mathematical Sociology 25, 163–177.
[13] Brandes, U., 2008. On variants of shortest-path betweenness centrality
and their generic computation. Social Networks 30, 136–145.
[14] Charkhgard, H., Subramanian, V., Silva, W., Das, T.K., 2018. An inte-
ger linear programming formulation for removing nodes in a network to
minimize the spread of influenza virus infections. Discrete Optimization
30, 144–167.
[15] Demˇsar, J., 2006. Statistical comparisons of classifiers over multiple
data sets. The Journal of Machine Learning Research 7, 1–30.
[16] Di Summa, M., Grosso, A., Locatelli, M., 2012. Branch and cut algo-
rithms for detecting critical nodes in undirected graphs. Computational
Optimization and Applications 53, 649–680.
33
[17] Ding, J., L¨u, Z., Li, C., Shen, L., Xu, L., Glover, F.W., 2019. A
two-individual based evolutionary algorithm for the flexible job shop
scheduling problem, in: The Thirty-Third AAAI Conference on Artifi-
cial Intelligence, AAAI 2019, AAAI Press. pp. 2262–2271.
[18] Divsalar, A., Vansteenwegen, P., S¨orensen, K., Cattrysse, D., 2014. A
memetic algorithm for the orienteering problem with hotel selection.
European Journal of Operational Research 237, 29–49.
[19] Doostmohammadian, M., Rabiee, H.R., Khan, U.A., 2020. Centrality-
based epidemic control in complex social networks. Social Network
Analysis and Mining 10, 32.
[20] Fan, C., Zeng, L., Sun, Y., Liu, Y., 2020. Finding key players in com-
plex networks through deep reinforcement learning. Nature Machine
Intelligence 2, 317–324.
[21] Hagberg, A., Swart, P., S Chult, D., 2008. Exploring network structure,
dynamics, and function using NetworkX. Technical Report. Los Alamos
National Lab.(LANL), Los Alamos, NM (United States). URL: https:
//www.osti.gov/servlets/purl/960616.
[22] Hooshmand, F., Mirarabrazi, F., MirHassani, S., 2020. Efficient ben-
ders decomposition for distance-based critical node detection problem.
Omega 93, 102037.
[23] Katz, L., 1953. A new status index derived from sociometric analysis.
Psychometrika 18, 39–43.
[24] Landherr, A., Friedl, B., Heidemann, J., 2010. A critical review of
centrality measures in social networks. Business & Information Systems
Engineering 2, 371–385.
[25] L´opez-Ib´a˜nez, M., Dubois-Lacoste, J., C´aceres, L.P., Birattari, M.,
St¨utzle, T., 2016. The irace package: Iterated racing for automatic
algorithm configuration. Operations Research Perspectives 3, 43–58.
[26] L¨u, Z., Hao, J., 2010. A memetic algorithm for graph coloring. Euro-
pean Journal of Operational Research 203, 241–250.
[27] Moalic, L., Gondran, A., 2018. Variations on memetic algorithms for
graph coloring problems. Journal of Heuristics 24, 1–24.
34
[28] Neri, F., Cotta, C., 2012. Memetic algorithms and memetic computing
optimization: A literature review. Swarm and Evolutionary Computa-
tion 2, 1–14.
[29] Pullan, W., 2015. Heuristic identification of critical nodes in sparse
real-world graphs. Journal of Heuristics 21, 577–598.
[30] Purevsuren, D., Cui, G., Qu, M., Win, N.H., 2017. Hybridization of
GRASP with exterior path relinking for identifying critical nodes in
graphs. IAENG International Journal of Computer Science 44, 157–
165.
[31] Ren, J., Hao, J.K., Wu, F., Fu, Z.H., 2023. An effective hybrid search
algorithm for the multiple traveling repairman problem with profits.
European Journal of Operational Research 304, 381–394.
[32] Salemi, H., Buchanan, A., 2022. Solving the distance-based critical
node problem. INFORMS Journal on Computing 34, 1309–1326.
[33] de San L´azaro, I.M., S´anchez-Oro, J., Duarte, A., 2021. Finding criti-
cal nodes in networks using variable neighborhood search, in: Proceed-
ings of Variable Neighborhood Search - 8th International Conference,
ICVNS 2021, Abu Dhabi, United Arab Emirates, March 21-25, 2021,
pp. 1–13.
[34] Ventresca, M., 2012. Global search algorithms using a combinatorial
unranking-based problem representation for the critical node detection
problem. Computers & Operations Research 39, 2763–2775.
[35] Ventresca, M., Aleman, D., 2013. Evaluation of strategies to mitigate
contagion spread using social network characteristics. Social Networks
35, 75–88.
[36] Ventresca, M., Aleman, D., 2014. A randomized algorithm with lo-
cal search for containment of pandemic disease spread. Computers &
Operations Research 48, 11–19.
[37] Ventresca, M., Aleman, D., 2015a. Efficiently identifying critical nodes
in large complex networks. Computational Social Networks 2, 1–16.
[38] Ventresca, M., Aleman, D., 2015b. Network robustness versus multi-
strategy sequential attack. Journal of Complex Networks 3, 126–146.
35
[39] Ventresca, M., Harrison, K.R., Ombuki-Berman, B.M., 2018. The bi-
objective critical node detection problem. European Journal of Opera-
tional Research 265, 895–908.
[40] Veremyev, A., Boginski, V., Pasiliao, E.L., 2014a. Exact identifica-
tion of critical nodes in sparse networks via new compact formulations.
Optimization Letters 8, 1245–1259.
[41] Veremyev, A., Prokopyev, O.A., Pasiliao, E.L., 2014b. An integer pro-
gramming framework for critical elements detection in graphs. Journal
of Combinatorial Optimization 28, 233–273.
[42] Veremyev, A., Prokopyev, O.A., Pasiliao, E.L., 2015. Critical nodes for
distance-based connectivity and related problems in graphs. Networks
66, 170–195.
[43] Wang, Z., Di, Y., 2022. Cluster expansion method for critical node
problem based on contraction mechanism in sparse graphs. IEICE
TRANSACTIONS on Information and Systems 105, 1135–1149.
[44] Zhang, Y., Chen, Q., Chen, B., Liu, J., Zheng, H., Yao, H., Zhang,
C., 2020. Identifying hotspots of sectors and supply chain paths for
electricity conservation in China. Journal of Cleaner Production 251,
119653.
[45] Zhao, Y., Cao, Y., Shi, X., Wang, S., Yang, H., Shi, L., Li, H., Zhang,
J., 2021. Critical transmission paths and nodes of carbon emissions in
electricity supply chain. Science of The Total Environment 755, 142530.
[46] Zhou, Y., Hao, J.K., Duval, B., 2017. Opposition-based memetic search
for the maximum diversity problem. IEEE Transactions on Evolution-
ary Computation 21, 731–745.
[47] Zhou, Y., Hao, J.K., Fu, Z.H., Wang, Z., Lai, X., 2021a. Variable
population memetic search: A case study on the critical node problem.
IEEE Transactions on Evolutionary Computation 25, 187–200.
[48] Zhou, Y., Hao, J.K., Glover, F., 2019. Memetic search for identifying
critical nodes in sparse graphs. IEEE Transactions on Cybernetics 49,
3699–3712.
[49] Zhou, Y., Kou, Y., Zhou, M., 2022. Bilevel memetic search approach
to the soft-clusteredvehicle routing problem. Transportation Science
URL: https://doi.org/10.1287/trsc.2022.1186.
36
[50] Zhou, Y., Wang, Z., Jin, Y., Fu, Z.H., 2021b. Late acceptance-based
heuristic algorithms for identifying critical nodes of weighted graphs.
Knowledge-Based Systems 211, 106562.
37
