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1
Memetic search for identifying critical nodes in
sparse graphs
Yangming Zhou, Jin-Kao Hao*, Fred Glover
IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2018.2848116
Abstract—Critical node problems involve finding a set of
critical nodes from a graph whose removal results in optimizing
a pre-defined measure over the residual graph. As useful models
for a variety of practical applications, these problems are
computational challenging. In this paper, we study the classic
Critical Node Problem (CNP) and introduce an effective memetic
algorithm for solving CNP. The proposed algorithm combines a
double backbone-based crossover operator (to generate promis-
ing offspring solutions), a component-based neighborhood search
procedure (to find high-quality local optima) and a rank-based
pool updating strategy (to guarantee a healthy population).
Extensive evaluations on 42 synthetic and real-world benchmark
instances show that the proposed algorithm discovers 24 new up-
per bounds and matches 15 previous best-known bounds. We also
demonstrate the relevance of our algorithm for effectively solving
a variant of the classic CNP, called the Cardinality-Constrained
Critical Node Problem (CC-CNP). Finally, we investigate the
usefulness of each key algorithmic component.
Index Terms—Heuristics, memetic search, critical node prob-
lems, sparse graph, complex networks.
I. INTRODUCTION
Given an undirected graph G= (V, E)with node set Vand
edge set E,critical node problems aim to identify a limited
number of nodes |S| ⊆ Vfrom Gsuch that their deletion
optimizes a predefined connectivity metric over the residual
graph G[V\S](i.e., the sub-graph of Ginduced by V\S).
The deleted nodes in Sare commonly called critical nodes.
CNPs have natural applications in a number of fields,
such as network security issues [30], epidemic control [36],
biological interaction networks [33], and social network anal-
ysis [8], [37]. For instance, in a social network, each node
corresponds to a person, edges represent interactions between
the individuals (e.g., friendship or collaboration), and critical
nodes correspond to the “key players” of the network (e.g.,
leaders of the organization or community) [8].
In this work, we are interested in the classic critical node
problem (defined in Section III and denoted as CNP here-
inafter) [5]. Informally, CNP involves finding a subset Sof
at most Knodes, whose removal minimizes the total number
of node pairs that are connected by a path in the residual
graph. CNP is known to be N P-hard on general graphs [5],
Y. Zhou is with the Department of Computer Science and Engineering,
East China University of Science and Technology, Shanghai, 200237, China
(e-mail: zhou.yangming@yahoo.com). J.K. Hao (Corresponding author) is
with the Department of Computer Science, Universit´
e d’Angers, 2 Boulevard
Lavoisier, 49045 Angers and also with the Institut Universitaire de France,
1 rue Descartes, 75231 Paris, France (e-mail: jin-kao.hao@univ-angers.fr). F.
Glover is with the University of Colorado, Leeds School of Business, Boulder,
CO, USA (e-mail: glover@opttek.com).
even if there are polynomially solvable special cases [1], [31].
The computational challenge and wide range of applications
of CNP have motivated a variety of solution approaches in the
literature, which are reviewed in Section II-A.
To enrich the set of solution methods for solving this
computationally challenging problem, we propose in this work
the first population-based memetic algorithm for CNP. The al-
gorithm combines an original component-based neighborhood
search procedure using an effective large component-based
node exchange strategy and a node weighting scheme (to find
high-quality local optima, Section IV-D), a double backbone-
based crossover operator (to generate promising new solutions,
Section IV-E) as well as a rank-based pool updating technique
(to maintain a healthy population, Section IV-F).
We assess the effectiveness of the proposed algorithm on 16
synthetic and 26 real-world benchmark instances commonly
used in the literature and especially report 24 improved best-
known results (new upper bounds). We show the generality of
the proposed approach by adapting the method to the related
cardinality-constrained critical node problem and report ad-
ditional 25 new best-known results. We also investigate each
key ingredient of the proposed approach to shed light on their
impacts on the performance of the method.
II. LITERATURE REVIEW
A. Algorithms for CNP
Due to their theoretical and practical significance, critical
node problems have attracted considerable research effort in
the last decade and several solution methods (both exact and
heuristic algorithms) have been proposed.
In general, exact algorithms are theoretically able to guar-
antee the optimality of the solutions they find. These methods
are particularly useful for special cases of graphs. For instance,
optimal polynomial-time dynamic programming algorithms
are proposed to solve two CNP variants for tree structures
and series-parallel graphs in [31]. A polynomial-time dynamic
programming algorithm for graphs with a bounded treewidth is
studied in [1]. For general graphs, a branch-and-cut algorithm
is proposed in [32] for CNP, using an integer linear program-
ming model with a non-polynomial number of constraints.
A more compact linear 0-1 formulation of the problem that
requires Θ(n2)constraints is recently developed and optimal
solutions for small networks are ascertained in [38], [39].
Given the N P-hard nature of critical node problems, it
is expected that any exact algorithm for finding the optimal
solution in the general case will require an exponential com-
putation time. Consequently, as for many N P -hard problems,
2
heuristics constitute a valuable alternative that are used to find
high-quality approximate solutions within a reasonable time
when exact methods are not usefully applicable. For instance,
an early heuristic starts with an independent set and uses
a greedy criterion to remove vertices [5]. Two other greedy
algorithms are presented in [27], [37], using advanced greedy
criteria for node selections. Moreover, several neighborhood-
based local search algorithms have been studied, like simulated
annealing [35], iterated local search and variable neighborhood
search [3]. These algorithms typically use the unconstrained
node exchange operation that swaps a node in Sagainst a
node in V\S, leading to a large and expensive neighborhood
of quadratic size. Population-based algorithms have also been
investigated, including path relinking with greedy randomized
adaptive search procedure [28] and genetic algorithm [4].
These algorithms maintain a pool of candidate solutions and
generate new solutions from existing ones by local changes,
crossover and random mutations.
B. Memetic algorithms
It is noteworthy that the popular memetic algorithms (MAs)
[23] have not been investigated for CNP. Indeed, as discussed
in several comprehensive surveys of memetic computation
(e.g., [10], [24], [25], [26]), MAs represent a highly effective
approach for solving difficult problems including numerous
N P -hard problems (e.g., graph coloring [21], graph partition
and biclique [7], [41], maximum diversity [40], quadratic
knapsack [11]) and routing problems [14].
MAs provide a general computational framework that
typically combines population-based approaches with
neighborhood-based local search. This framework offers
interesting opportunities for designing search algorithms
able to ensure a desirable balance between exploration and
exploitation of large search spaces [10], [19], [25], [26]. To
further improve the MA framework, interesting extensions
incorporating learning schemes have been proposed in the
literature. For example, [14] presents an evolutionary memetic
computing paradigm which is capable of learning and evolving
a knowledge meme that traverses different but related problem
domains. [15] proposes a memetic computational paradigm
based on evolutionary optimization + transfer learning for
search, which mimics human problem-solving by transferring
structured knowledge learned from solving related problems
in the form of memes. In [44], opposition-based learning is
introduced into memetic search for solving the maximum
diversity problem.
This work presents the first memetic algorithm dedicated to
the classic critical node problem. As demonstrated with our
computational results, the proposed algorithm competes very
favorably with the existing CNP approaches.
III. PROB LE M DE SC RI PT IO N AN D NOTATI ON
Critical node problems in a graph G= (V, E )aim to
remove a “limited” subset of nodes S⊆Vin order to
optimize a pre-defined connectivity measure over the residual
graph G[V\S]. Once the critical nodes have been removed,
the residual graph G[V\S]can be represented by a set of
disjoint connected components H={C1,C2,...,CT}, where
a connected component Ciis a set of nodes such that all nodes
in this set are mutually connected (reachable by a path), and
there is no edge between any two connected components.
According to the particular interests, different connectivity
measures have been studied, which can be divided into three
categories.
(i) To optimize the pair-wise connectivity, i.e., the number of
pairs of nodes connected by a path in the residual graph
[1], [2], [3], [4], [5], [13], [27], [28], [37], [39].
(ii) To optimize the size of the largest connected component
in the residual graph [4], [27], [31], [39].
(iii) To optimize the number of connected components in the
residual graph [31], [39], [4].
One notices that most studies in the literature have focused
on the classic Critical Node Problem (denoted as CNP), which
involves minimizing the pair-wise connectivity measure [5]
and belongs to the first category mentioned above. Formally,
given a graph G= (V, E)and an integer K, CNP is to identify
a subset S⊆Vsuch that |S|6K, whose removal minimizes
the following pair-wise connectivity objective f(S):
f(S) =
T
X
i=1 |Ci|
2(1)
where Tis the total number of connected components Ci
in the residual graph G[V\S].
v7
v
6
v1
v
3
v
4
v5
(a)
v
8
v7
v
6
v1
v
3
v
4
v5
v
8
(b)
v2
v2
Fig. 1. A CNP example. (a) An undirected graph G with 8 nodes and K= 2.
(b) A candidate solution for the instance (i.e., a subset S={v4, v7}) and its
cost f(S)can be computed as 2
2+4
2= 7.
Fig. 1 shows a CNP instance (i.e., the graph of Fig. 1(a)
with eight nodes and K= 2) and a candidate solution
S={v4, v7}(Fig. 1(b)), whose removal from the graph leads
to two connected components induced by {v1, v2, v3, v8}and
{v5, v6}with an objective value f(S) = 7.
CNP can be also considered as a problem of maximally
fragmenting a graph and simultaneously minimizing the vari-
ance among the sizes of connected components in the residual
graph. In other words, the resulting residual graph should be
composed of a relatively large number of connected compo-
nents while each connected component has a similar size [37].
In this paper, we focus on solving this classic critical node
problem. In Section VI, we also show the applicability of our
memetic algorithm to solve an important variant of CNP, i.e.,
the Cardinality-Constrained Critical Node Problem (CC-CNP)
[6], which falls into the second category mentioned above.
3
IV. THE P ROPOSED MEMETIC APPROACH FOR CNP
Given a CNP instance defined by G= (V, E)and K∈Z+,
let S(|S|< K) be a candidate solution of G. It is easy to
verify that any solution S0=S∪ {u}with one more node
u∈V\Sis a feasible solution no worse than S(i.e., f(S0)6
f(S)). Indeed, if uis an isolated node in G[V\S], adding u
to Shas no impact on the objective value. Otherwise, let u
belong to a connected component, say Ci, moving ufrom Ci
to Sreduces the size of Ciby one or disconnects Ciinto two
or more new components. In both cases, we have f(S0)6
f(S). Since Kis the largest possible value for a solution to
be feasible, wa can safely consider only candidate solutions S
with exactly Kselected elements.
A. Solution representation and evaluation
Given a graph G= (V, E)and an integer K, any subset S⊂
Vof Knodes is a feasible solution and can be represented
as S={vS(1), vS(2) , . . . , vS(K)}(16S(i)6=S(j)6|V|
for all i6=j) where S(l)(16l6K) is the index of a
selected node in S. Therefore, the solution space Ωcontains
all possible subsets S⊆Vsuch that |S|=K.
According to Equation (1), the corresponding objective
value f(S)of a feasible solution Sis the total number of
node pairs still connected by a path in the residual graph
G[V\S].f(S)can be computed in O(|V|+|E|)time with
a modified depth-first search algorithm by identifying the
connected components of a graph [20] as follows. One finds
the connected components of a graph by performing the depth-
first search on each connected component. Each new node
visited is marked. When no more nodes can be reached along
the edges from the marked nodes, a connected component is
found. Then, an unvisited node is selected, and the process is
repeated until the entire graph is explored.
B. General scheme
The proposed MACNP algorithm is composed of four
main procedures: a population initialization procedure, a
component-based neighborhood search procedure, a double
backbone-based crossover procedure and a rank-based pool
updating procedure. MACNP starts from a set of distinct elite
individuals which are obtained by the population initialization
procedure (Section IV-C). At each generation, an offspring
solution is generated by the double backbone-based crossover
procedure (Section IV-E). This offspring solution is further
improved by the component-based neighborhood search pro-
cedure (Section IV-D) and then considered for acceptance by
the rank-based pool updating procedure (Section IV-F). The
process is repeated until a stopping condition (e.g., time limit)
is satisfied. We show the flowchart and pseudo-code of the
MACNP algorithm in Fig. 2 and Algorithm 1. We present the
key procedures in the following sections.
C. Population initialization
Our MACNP algorithm starts its search with an initial
population composed of diverse and high-quality solutions.
To construct such a population, we first generate randomly
Generate an initial population P of size p;
identify the best individual with minimum
objective function value as S*;
set the running time t = 0;
Randomly select two parent
solutions from P
Generate an offspring solution S' by
performing double backbone crossover
on two chosen solutions
Improve S' by a component
based neighborhood search
Update the population P based on
a rank based pool updating strategy;
update the best solution S*
no
Output the best solution
yes
t<tmax ?
Fig. 2. Flowchart of the proposed MACNP algorithm
Algorithm 1: Pseudo-code of MACNP algorithm.
Input: An undirected graph G = (V,E) and an integer K
Output: The best solution S∗found so far
1begin
2// build an initial population, Section IV-C
3P={S1, S2,...,Sp} ← PoolInitialize();
4S∗←arg min{f(Si) : i= 1,2,...,p};
5while a stopping condition is not reached do
6randomly select two parents Siand Sjfrom P;
7// generate an offspring by crossover, Section IV-E
8S0←DoubleBackboneBasedCrossover(Si, Sj);
9// perform a local search, Section IV-D
10 S0←ComponentBasedNeighborhoodSearch(S0);
11 if f(S0)< f(S∗)then
12 S∗←S0;
13 // insert or discard the improved solution, Section
IV-F
14 P←RankBasedPoolUpdating (P, S0)
15 return the best solution found S∗;
a feasible solution (i.e., any set of at most Knodes), and
then we improve it by the component-based neighborhood
search procedure described in Section IV-D. We insert the
improved solution into the population if it is different from the
existing individuals of the population. Otherwise, we modify
the improved solution with the exchange operation until it
becomes different from all existing individuals before inserting
it into the population. We repeat the procedure ptimes to fill
the population with pdistinct solutions.
D. Component-based neighborhood search
To ensure an effective local optimization, MACNP employs
a fast and effective Component-Based Neighborhood Search
(CBNS) procedure (see Algorithm 2). CBNS integrates two
key techniques, i.e., a large component-based node exchange
strategy and a node weighting technique.
4
Algorithm 2: Component-based neighborhood search
Input: A solution S, number of idle iterations
MaxIdleIters.
Output: The best solution S∗found so far
1S∗←S;
2iter ←0,idle iter ←0;
3while idle iter < MaxIdleIters do
4select a large component cat random;
5remove a node vfrom component cwith the node
weighting scheme;
6S←S∪ {v};
7u←arg minw∈S{f(S\ {w})−f(S)};
8S←S\ {u};
9if f(S)< f(S∗)then
10 S∗←S;
11 idle iter ←0;
12 else
13 idle iter ←idle iter + 1;
14 iter ←iter + 1;
15 return the best solution found;
1) Component-based neighborhood structure: The perfor-
mance of a local search procedure greatly depends on its
neighborhood structure for candidate solution exploration. A
traditional neighborhood for CNP is defined by the conven-
tional exchange operator which swaps a node u∈Swith a
node v∈V\S[2], [28], [35]. For a given solution, this neigh-
borhood yields O(K(|V|−K)) neighbor solutions. To evaluate
a neighbor solution, no incremental technique is known and a
full computation from scratch is required by running the mod-
ified depth-first search algorithm of complexity O(|V|+|E|)
[20]. Therefore, examining the whole neighborhood requires
a time of O(K(|V| − K)(|V|+|E|)), which becomes too
expensive when many local search iterations are performed
(which is usually the case).
Recently, two other refined neighborhoods have been pro-
posed in [3]. For a given u∈S, the first neighborhood aims to
directly determine the node v= arg max{f(S)−f((S\{u})∪
{v0})},∀v0∈V\Sso that swapping uand vdisconnects
the graph as much as possible. For v∈V\S, the second
neighborhood tries to identify the node u←arg min{f((S∪
{v})\{u0})−f(S)},∀u0∈S. The computational complexity
of examining these neighborhoods is O(K(|V|+|E|)) and
O((|V|−K)(|V|+|E|+K×degree(G))) respectively, where
degree(G)is the maximum node degree in G. Even if these
neighborhoods are more computationally efficient compared
to the traditional swap neighborhood, they are still expensive
to explore within a local search algorithm.
To overcome the limitation, we design an alternative
component-based neighborhood, which is both smaller in size
and more focused with respect to the optimization objective.
Recall that CNP involves fragmenting the graph in order to
minimize the number of connected node pairs in the residual
graph G[V\S]. This can be achieved by fragmenting the
largest connected components in G[V\S]in order to obtain
more homogeneous components, which helps to minimize
the number of node pairs still connected. As a result, when
exchanging a node u∈Swith a node v∈V\S, it is preferable
to consider v∈V\Sfrom a large component (see Definition
(1) below) instead of a small component. Let Lbe a predefined
threshold to qualify large components. We consider only a
subset of nodes Z⊂V\Ssuch that Z=∪|Ci|>LCias can-
didate nodes for exchanges. Consequently, the neighborhood
size is reduced to K|Z|, which is generally far smaller than
K(|V| − K)for reasonable Lvalues.
Definition 1 (large component): A connected component in
the residual graph G[V\S]qualifies as a large component if
the number of its nodes is greater than the predefined threshold
L= (max nc+min nc)/2, where max ncand min ncare
respectively the number of nodes in the largest and smallest
connected components in the residual graph G[V\S].
v7
v
6
v1
v
3
v
4
v5
(a)
v
8
v7
v
6
v1
v
3
v
4
v5
v
8
(b)
v2
v2
Fig. 3. An illustrative example of component-based neighborhood structure.
(a) A candidate solution S. (b) An improved neighbor solution S0of S.
Fig. 3(a) shows a CNP solution S={v4, v7}(K= 2) with
f(S)=7. After deleting v4and v7from the original graph G,
the resulting residual graph is composed of two components
C1={v5, v6}and C2={v1, v2, v3, v8}. Our component-
based neighborhood always swaps a node u∈Swith a
node selected from a predefined large component (instead
of a random component). Based on the definition (1), we
have L= (4 + 2)/2 = 3. Therefore we only consider the
swapping operations between Sand a large component C2
(|C2|> L = 3), thus reducing the neighbor solutions to be
considered from 12 to 8. Fig. 1(b) shows an improved neighbor
solution S0with f(S0) = 0 + 2
2+3
2= 4.
2) Two-phase node exchange strategy: To further reduce
the size of the above component-based neighborhood, we
employ a two-phase node exchange strategy which relies on
neighborhood decomposition [17], [29], [44]. This strategy
breaks an exchange operation on a node pair into two distinct
phases: a “removal phase” removes a node from the residual
graph and an “add phase” adds a node to the residual graph.
In our case, we first randomly select a large component and
removes a node vfrom the selected component with the
node weighting scheme (see below). For the node u∈Sto
be moved to the residual graph, we select the node which
minimally deteriorates the objective function. With the help
of this exchange strategy, the computational effort required
to examine the candidate solutions greatly decreases. For
instance, for the graph ‘BA1000’ with 1000 vertices and
K= 75, using the large component-based two-phase node
exchange strategy reduces the number of candidate solutions
from (1000 −75) ×75 = 69375 to 75 69375.
For such a two-phase exchange, selecting a node vto
remove from G[V\S]can be performed in O(T+nbrv), where
Tis the number of connected components in G[V\S]and
5
nbrvis the length of the adjacency list of node v. Once a node
is added into S, we need to remove a node ufrom S. We select
uthat causes the minimum increase in the objective function.
The evaluation of the increase in the objective function for
each node in Sis performed by scanning the adjacency list
of the node to determine if its removal will re-connect some
existing components to form a large component (as the node
being added back must be in the same component as all
its neighboring nodes in the residual graph G[V\S]). This
operation requires time O(K×nbru)where nbruis the length
of the adjacency list of node u.
3) Node weighting scheme: The node weighting technique
[34], [9] is the second useful technique we adopted in the
component-based neighborhood search. Our node weighting
scheme works as follows. Each node of a large component is
associated with a non-negative integer as its weight, initialized
to 0. At each step, we randomly select a component Ciamong
the large connected components, and select the node vin Ci
with the largest weight (breaking ties in favor of the node
with the largest degree) to move to S. Simultaneously, the
weights of the remaining nodes in Ciare increased by one.
Additionally, when a node v∈ Ciis exchanged with a node
u∈S, we set the weight of uto 0.
With the node weighting scheme, the “hard to remove”
nodes will have larger weights, and thus have a higher chance
to be considered for removal from the component in the
following iterations. This technique helps the search to escape
from potential local optima. Our weighting scheme follows
the general penalty idea for constraint satisfaction problems,
which was first used in this setting in the breakout method
[22]. We note that this scheme is also an instance of a
tabu search frequency-based memory [16]. To the best of our
knowledge, it is the first time that a node weight learning
technique is applied to a heuristic procedure for CNP.
E. Double backbone-based crossover
Crossover is another important ingredient of the MACNP
algorithm. It should be noted that the meaning of “crossover”
has changed from the genetic conception adopted in the early
formulation of memetic algorithms. The modern conception
embraces the principle of structured combinations introduced
in [18], where solutions are combined by domain specific
heuristics that map them into new solutions faithful to the
structure of the problems considered. A similar evolution in
the notion of crossover has been occurring within genetic al-
gorithms to incorporate the notion of structured combinations,
although often incompletely. As observed in [19], a successful
crossover should be able to generate promising offspring
solutions by inheriting good properties of the parents and
introducing useful new features, while respecting the domain
specific structure of the problem context. The concept of
backbone has been used to design some successful crossover
operators for subset selection problems [40], [44]. Since CNP
is a typical subset selection problem, we adopt the backbone
idea and design a double backbone-based crossover operator
to create structured combinations as follows.
Let S1and S2be two solutions of CNP. According to
S1and S2, we divide the set of elements Vinto three
subsets denoted by common elements, exclusive elements and
excluding elements, as shown in Definitions 2-4.
Definition 2 (common elements): The set of common
elements XAis the set of elements of Vshared by S1and
S2, i.e., XA=S1∩S2.
Definition 3 (exclusive elements): The set of exclusive
elements XBis the set of elements of Vwhich are in either
S1or S2and not in XA, i.e., XB= (S1∪S2)\(S1∩S2)
(the symmetric difference of S1and S2).
Definition 4 (excluding elements): The set of excluding
elements XCis the set of elements of Vwhich are not
included in S1and S2, i.e., XC=V\(S1∪S2).
From two parent solutions S1and S2randomly selected
from the population P, an offspring solution S0is constructed
in three phases: (i) create a partial solution by inheriting all
common elements (i.e., the first backbone), i.e., S0←XA;
(ii) add exclusive elements (i.e., the second backbone) into
the partial solution in a probabilistic way. That is, for each
exclusive element, we add it into S0with probability sp0(0<
sp0<1); (iii) repair the partial solution structurally until a
feasible solution is achieved. Specifically, if |S0|< K, we
randomly add some excluding elements of XCto S0from a
random large connected component in the residual graph. If
|S0|> K, we greedily remove some elements from S0until
|S0|=K. The elements added in the first two phases form the
whole backbone of the parent solutions. Therefore, the double
backbones are composed of |XA|common elements and about
sp0× |XB|exclusive elements. Note that sp0, which could be
tuned statically or dynamically (e.g., as a function of |XB|,
|XA|and K), is used to influence the diversity of the offspring
solution by including excluding elements.
The proposed crossover shares ideas related to those of
the crossovers proposed in [40], [44] by directly inheriting
common elements from the parent solutions. However, to
complement the partial solution, our crossover favors the
transmission of exclusive elements from the parent solutions
over the excluding elements that are not in the parent solutions.
As such, our crossover is less destructive and has a stronger
intensification effect (see also Section VII-C for a comparative
study). It is worth noting that the proposed crossover is
also different from the crossover for critical node problems
introduced in [4] that uses set union as its key operation.
F. Rank-based pool updating
Each offspring solution is submitted for improvement by the
component-based neighborhood search procedure presented in
Section IV-D. Then we use a rank-based pool updating strategy
to decide whether the improved offspring solution S0should
be accepted in the population, as shown in Algorithm 3). This
pool updating strategy resorts to a score function to evaluate
each individual, which not only considers the quality of the
offspring but also its average distance to other individuals in
the population. This strategy is inspired by the population
management strategies presented in [11], [21], [44].
First, we temporarily insert S0into the population (line 1
of Algorithm 3). Then we evaluate all the solutions of the
population according to the score function [44] (lines 2-3 of
6
Algorithm 3) and identify the worst solution Sw(line 4 of
Algorithm 3). Finally, if S0is different from Sw, we replace
Swby S0. Otherwise, we discard S0(lines 5-6 of Algorithm
3).
Algorithm 3: Rank-based pool updating strategy
Input: A population Pof size pand an improved solution
S0
Output: An updated population P
1P0←P∪ {S0};
2for i= 0,1,...,p do
3evaluate solution Siaccording to the score function;
4identify the worst solution Swin population P0i.e.,
w←maxj∈{0,1,...,p}Score(Sj, P 0);
5if w6= 0 then
6replace Swwith S0, i.e., P←P0\ {Sw};
7return the updated population P;
G. Computational complexity of MACNP
To analyze the computational complexity of the proposed
MACNP algorithm, we consider the main steps in one gener-
ation in the main loop of Algorithm 1.
As displayed in Algorithm 1, each generation of MACNP
performs four subroutines: parent selection, double backbone-
based crossover, component-based neighborhood search and
rank-based pool updating. The parent selection procedure takes
time O(1). The double backbone-based crossover operator
can be realized in O(K(|V|+|E|)). The component-based
neighborhood search is bounded in time by O(K(|V|+
|E|)MaxIters, where MaxIters is the total number of
iterations at each neighborhood search. The rank-based pool
updating can be achieved in time O(p(K2+p)), where p
is the population size. Hence, for each generation, the total
complexity of MACNP is O(K(|V|+|E|)MaxI ters).
V. COMPUTATIONAL STUDIES
This section presents computational studies to evaluate the
performance of our MACNP algorithm and compare it with
state-of-the-art algorithms.
A. Benchmark instances
Our computational studies were based on two popular
benchmark sets composed of 42 graphs1.
Synthetic benchmark set [35] contains 16 instances (with
235 to 5000 nodes) classified into four categories: Barabasi-
Albert (BA) graphs, Erdos-Renyi (ER) graphs, Forest-Fire
(FF) graphs and Watts-Strogatz (WS) graphs.
Real-world benchmark set [4] consists of 26 real-world
graphs (with 121 to 23133 nodes) from various practical
applications in areas like biology, electronics, transportation
and complex networks.
1Available at http://www.di.unito.it/∼aringhie/cnp.html
B. Experimental settings
The proposed MACNP algorithm2was implemented in
C++ and complied with gcc 4.1.2 and flag ‘-O3’. All the
experiments were carried out on a computer equipped with
an Intel E5-2670 processor with 2.5 GHz and 2 GB RAM
operating under the Linux system. Running the well-known
DIMACS machine benchmark procedure dfmax.c3on our
machine requires 0.19, 1.17 and 4.54 seconds to solve the
benchmark graphs r300.5, r400.5 and r500.5 respectively.
TABLE I
THE PAR AM ETE R SE TTI NGS O F TH E PRO POS ED MACNP ALGORITHM.
Parameter description value section
ppopulation size 20 IV-C
MaxI dleI ters maximum number of idle iterations in CBNS 1000 IV-D
sp0selection probability 0.85 IV-E
1) Parameter tuning: Like previous CNP algorithms and
most heuristic algorithms, MACNP requires some parameters
(see Table I). Parameter pis the population size (Section IV-C)
and is set to p= 20 as suggested in [21] [40] and [44].
Parameter MaxIdleIters is the maximum allowable number
of idle iterations, which is used as the stopping condition of
the component-based neighborhood search (Section IV-D). sp0
determines the probability of inheriting an exclusive element
during the crossover operation (Section IV-E). Our analyses
indicate that MaxIdleIters and sp0are sensitive parameters
while this is not the case for p. To tune M axIdleIters and
sp0, we followed the common practice in the heuristic litera-
ture by testing a limited number of parameter configurations
on a small sample of problem instances as follows.
selection probability sp0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
solution quality
×106
1
1.02
1.04
selection probability sp0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
time (seconds)
50
100
150
200
Fig. 4. Box plots corresponding to different values of sp0∈[0,1.0] in terms
of solution quality (above sub-figure) and computation time (bottom sub-
figure). X-axis indicates each sp0value and Y-axis shows the performance.
To avoid over-fitting, we used 9 (out of 42) representative
instances (BA5000, ER941, FF500, WS250, TreniR, open-
2Our codes and computational results will be made available at http://www.
info.univ-angers.fr/∼hao/cnps.html
3dfmax: ftp://dimacs.rutgers.edu/pub/dsj/clique
7
flights, H3000a, H4000, OClinks) to cover both synthetic and
real-world instances with different sizes and variable levels of
difficulty. To identify a suitable value for a given parameter,
we evaluated the performance of MACNP by varying the pa-
rameter value within a reasonable range, while fixing the other
parameters to the default values of Table I. For each parameter
configuration, we ran MACNP 30 times to solve each sample
instance, each run being limited to 100 generations.
We take the parameter sp0as an example. Fig. 4 shows
the box plots with eleven values sp0∈[0,1.0]. We observe
that large values (sp0>0.8) yield better results than small
ones (sp0≤0.8). We also see that MACNP with sp0= 0.9
performs the best in terms of computation time. Meanwhile,
given that sp0influences the diversity of offspring solutions,
we adopted sp0= 0.85 in this work to make a reasonable
compromise between solution quality, computation time and
offspring diversity. In passing, since our double backbone
crossover reduces to the traditional backbone crossover when
sp0= 0.0(Section IV-E), this experiment also demonstrates
the advantage of the proposed crossover over the traditional
backbone crossover. To tune the parameter MaxI dleI ters,
we used the same procedure to identify 1000 as its default
value. Finally, note that, like many search algorithms, finding
the best parameter setting of our algorithm is not a trivial task
and may be time consuming. Possible solutions to cope with
this problem in general would be to reduce the number of
required parameters, define self-tunable parameters and make
use of automatic parameter tuning techniques.
For our computational studies, we used consistently the
parameter values of Table I to run our MACNP algorithm.
Notice that this parameter setting can be considered as the
default setting of MACNP and can conveniently be adopted
by a user. Meanwhile, users can also tune some parameters
in order to better solve their particular problem instances.
Given the stochastic nature of the algorithm, MACNP was
independently executed 30 times on each benchmark instance
like [35]. Following the standard practice for solving CNPs
[3], [4], [28], [42], we adopted a time limit (tmax = 3600
seconds) as the stopping condition of each run.
2) Statistical test: To analyze the experimental results, we
resort to the well-known two-tailed sign test [12] to check the
significant difference on each comparison indicator between
the compared algorithms. When two algorithms are compared,
the corresponding null-hypothesis is that the algorithms are
equivalent. The null-hypothesis is accepted if and only if each
algorithm wins on approximately X/2out of Xinstances.
Since tied matches support the null-hypothesis, we split them
evenly between the two compared algorithms, i.e., each one
receives the value 0.5. At a significance level of 0.05, the
corresponding Critical Values (CV) of the two-tailed sign test
are respectively CV 16
0.05 = 12 and CV 26
0.05 = 18 when the
number of instances in each benchmark is X= 16 and
X= 26. Consequently, Algorithm A is significantly better
than algorithm B if A wins at least CV X
0.05 instances for a
benchmark of Xinstances.
TABLE II
PERFORMANCE OF MACNP ON SYNTHETIC AND REAL-WOR LD
BENCHMARKS.
Instance K K BV [32] ∆fbest ∆favg tavg #exch
BA500 50 195∗0 0.0 0.0 5.8×102
BA1000 75 558∗0 0.0 0.3 6.4×103
BA2500 100 3704∗0 0.0 0.7 8.6×103
BA5000 150 10196∗0 0.0 6.5 3.5×104
ER235 50 295∗0 0.0 7.1 2.0×105
ER466 80 1524 0 0.0 28.5 9.5×105
ER941 140 5012 0 2.1 458.5 1.2×107
ER2344 200 959500 -57002 -37160.5 2284.8 1.5×107
FF250 50 194∗0 0.0 0.0 2.3×103
FF500 110 257∗0 0.0 0.4 8.7×103
FF1000 150 1260∗0 0.0 84.9 2.6×105
FF2000 200 4545∗0 0.7 107.6 3.3×105
WS250 70 3101 -18 -11.6 1140.5 6.1×107
WS500 125 2078 -6 4.6 179.3 2.4×106
WS1000 200 113638 -3831 10044.6 2675.0 1.8×107
WS1500 265 13167 -69 88.1 1012.2 7.3×106
Bovine 3 268 0 0.0 0.0 6.8×101
Circuit 25 2099 0 0.0 0.2 1.7×104
Ecoli 15 806 0 0.0 0.0 6.2×102
USAir97 33 4336 0 0.0 756.5 2.5×106
humanDi 52 1115 0 0.0 0.6 1.7×104
TreniR 26 920 -2 -2.0 0.3 2.2×104
EU fli 119 349927 -1659 1730.0 232.6 1.1×105
openfli 186 28671 -1829 33.3 2093.7 2.9×106
yeast1 202 1414 -2 -2.0 21.7 1.0×105
H1000 100 328817 -22468 -18190.5 2137.5 2.5×107
H2000 200 1309063 -65204 -45567.4 2861.9 1.2×107
H3000a 300 3005183 -160790 -120401.3 3280.7 1.1×107
H3000b 300 2993393 -152123 -108306.0 3252.9 1.2×107
H3000c 300 2975213 -136784 -105864.5 3307.5 1.1×107
H3000d 300 2988605 -157294 -96042.3 3250.9 1.1×107
H3000e 300 3001078 -153169 -113552.3 3437.4 1.2×107
H4000 400 5403572 -250595 -136196.5 2907.0 6.6×106
H5000 500 8411789 -439264 -316976.4.7 3226.6 6.3×106
powergr 494 16099 -237 -197.5 1286.4 1.8×106
OClinks 190 614504 -2201 40.0 584.6 4.5×105
faceboo 404 420334 222828 319102.6 2978.5 2.2×106
grqc 524 13736 -140 -106.8 871.8 9.2×105
hepth 988 114382 -7985 -4726.4 3442.0 3.1×106
hepph 1201 7336826 1291861 2033389.3 3376.3 1.4×106
astroph 1877 54517114 7551852 8030784.1 1911.4 3.9×105
condmat 2313 2298596 7155765 7763211.8 1779.5 4.9×105
∗Optimal results obtained by exact algorithm [32] within 5 days.
C. Performance of the MACNP algorithm
Table II shows the computational results for MACNP on
the synthetic and real-world benchmarks under the time limit
tmax = 3600 seconds. Columns 1-3 respectively describe
for each instance its name (Instance), the number of critical
nodes (K) and the known best value (KBV ) reported in [32].
Columns 4-8 report the detailed results of MACNP, including
the difference between the best objective value fbest and its
known best value KBV (i.e., ∆fbest =fbest −K BV ), the
difference between the average objective value favg and KBV
(i.e., ∆favg =favg −KBV ), the average time in seconds to
attain the objective value (tavg) and the average number of
node exchanges to achieve the objective value (#exch).
From Table II, we observe that MACNP attains the best
objective values for all 16 synthetic instances (first part of
the table) and finds in particular 5 new upper bounds (see
∆fbest <0). For instances ER2344 and WS250, our average
objective values are also better than the previously known
best values (see ∆favg <0). To the best of our knowledge,
8
our MACNP algorithm is the first heuristic which reaches the
optimal solution 4545 of FF2000 within the short computation
time of 107.6 seconds, which is far less than 5 days by the
branch-and-cut algorithm [32] (as reported in [3]). For the
real-world benchmark (second part of Table II), MACNP also
shows a highly competitive performance. It matches 5 known
best results and finds improved upper bounds for 17 out of the
26 instances. Also, the average objective value achieved by our
MACNP algorithm is better than the previous upper bound for
14 instances (see ∆favg <0in Table II). However, MACNP
failed to attain the known best values for four instances
(OClinks, hepph, astroph, and condmat) within the time limit
of tmax = 3600 seconds. Indeed, this time limit is too short for
the population-based MACNP algorithm to converge. Note that
in [4], a large time limit of tmax = 16000 seconds was used.
When we re-ran our MACNP algorithm under this condition,
MACNP managed to find better solutions, including two new
upper bounds for hepph and astroph (see results displayed in
italic format in Table IV). This experiment demonstrates the
effectiveness of our MACNP algorithm for solving the CNP
on both the synthetic and real-world benchmarks.
D. Comparison with the state-of-the-art algorithms
To further assess the performance of our MACNP algo-
rithm, we carried out detailed comparisons between MACNP
and 7 state-of-the-art reference algorithms: dynamic restarting
greedy algorithms (Greedy3d and Greedy4d) [2], iterated
local search (ILS) [3], variable neighborhood search (VNS)
[3], genetic algorithm (GA) [4], multi-start greedy algorithm
(CNA1) [27] and fast heuristic (FastCNP) [42].
Since the source codes of CNA1 [27] and FastCNP [42] are
available to us, we first present a detailed comparison between
MACNP and these two reference algorithms. It is worth noting
that the FastCNP algorithm used in our experiment is an
improved version of the initial FastCNP algorithm proposed
in [42], by additionally integrating the node weighting scheme
(Section IV-D3) in the neighborhood search procedure. To
make a fair comparison, the three algorithms were run on
our platform with the same time limit tmax = 3600 seconds,
and we ran each algorithm 30 times to solve each instance.
In addition to tmax, we used the default parameter values of
CNA1 and FastCNP: the number of nodes to remove in the
partial restart of CNA1 [27] and the maximum number of
idle iterations in local search and the perturbation strength for
FastCNP [42]. As for MACNP, the default parameter values
for CNA1 and FastCNP have been obtained based on sample
instances of the same CNP benchmark sets.
The comparative results are shown in Table III where the
first column provides the name of each instance (Instance),
Columns 2-5 report the results of the CNA1 algorithm, includ-
ing the best objective value (fbest) found during 30 runs, the
average objective value (favg ), the average time in seconds
(tavg ) and the average number of steps (#exch) needed to
achieve the best objective value at each run. Correspondingly,
columns 6-9 and columns 10-13 respectively present the
results of FastCNP and MACNP. The best values of the
compared results are in bold, and when the same best objective
values are achieved, fewer node exchanges are underlined
(which indicates a better performance in terms of computation
efficiency). In addition, we give the number of instances
(wins) for which our MACNP algorithm obtained a better
performance (i.e., fbest and favg ) compared to each competing
algorithm. The win values for indicators tavg and #exch are
meaningless and are marked by ‘*’.
Table III indicates that our MACNP algorithm significantly
outperforms CNA1 and FastCNP, achieving the best objective
values for 38 out of the 42 instances, and the best average
objective values for 37 instances. For the synthetic benchmark,
MACNP is significantly better than CNA1 in terms of the best
objective value, winning 12.5 instances (i.e., 12.5> CV 16
0.05 =
12). Compared to FastCNP, MACNP is also very competitive
and wins 10.5 instances, which is slightly smaller than the
critical value CV 16
0.05 = 12. As to the average objective value,
MACNP significantly outperforms both CNA1 and FastCNP
by winning 13 instances. For the real-world benchmark,
MACNP again proves to be significantly better than CNA1
and FastCNP both in terms of the best objective value and the
average objective value. Moreover, for the 14 instances where
all three algorithms attain the same best objective values, our
MACNP algorithm needs the least number of node exchanges
to reach its results (see values underlined).
We also compared our MACNP algorithm with five addi-
tional algorithms reported in the literature: dynamic restarting
greedy algorithms (Greedy3d and Greedy4d) [2], iterated local
search (ILS) [3], variable neighborhood search (VNS) [3],
and genetic algorithm (GA) [4]. Since the source code of
these reference algorithms is not available, we used their
best results reported in the corresponding papers. These five
algorithms have been recently evaluated on the same platform
(i.e., an HP ProLiant DL585 G6 server with two 2.1 GHz
AMD Opteron 8425HE processors and 16 GB of RAM) [3],
[4], which is slower than our machine with a factor 0.84
according to the Standard Performance Evaluation Corporation
(www.spec.org). However, their results were obtained under
different time limits: tmax ∈(7200,10000] for most of the
synthetic instances and tmax ∈[3000,16000] for most of the
real-world instances. The comparative results of MACNP with
the seven state-of-the-art heuristic algorithms on the synthetic
and real-world benchmarks are summarized in Table IV. Note
that the result of “Best ILS” for each instance is the best result
among 6 ILS variants, and “Best VNS’ corresponds to the best
result among all 24 VNS variants [3].
Table IV shows that our MACNP algorithm attains the
best results for all instances. Specifically, MACNP finds 6
new upper bounds and reaches the best objective values for
the remaining 10 instances. MACNP is significantly better
than Greedy3d, Greedy4d, Best VNS, Best ILS, GA, and
CNA1, respectively winning 15.0, 15.0, 12.5, 15.5, 12.0,
12.5 instances. Compared to FastCNP, MACNP wins 10.5
instances, which is just slightly smaller than the critical value
CV 16
0.05 = 12. These observations indicate that compared to the
state-of-the-art algorithms, our MACNP algorithm is highly
competitive for solving the synthetic instances.
Similar observations are found on the real-world benchmark
in Table IV. MACNP significantly outperforms the reference
9
TABLE III
COMPARATIVE PERFORMANCE OF THE PROPOSED MACNP WITH CNA1 AN D FAST CNP ON SYNTHETIC AND REAL-WORLD BENCHMARKS.
CNA1 FastCNP?MACNP
Instance fbest favg tavg #exch fbest favg tavg #exch fbest fav g tavg #exch
BA500 195 195.0 2.5 1.4×105195 195.0 <0.1 4.1×103195 195.0 <0.1 5.8×102
BA1000 558 558.7 5.4 1.1×105558 558.0 29.8 1.6×106558 558.0 0.3 6.4×103
BA2500 3704 3704.0 1.7 1.4×1043704 3710.6 649.9 1.7×1073704 3704.0 0.7 8.6×103
BA5000 10196 10196.0 103.7 3.8×10510196 10201.4 104.9 1.5×10610196 10196.0 6.5 3.5×104
ER235 295 295.0 6.8 1.0×106295 295.0 11.7 4.2×106295 295.0 7.1 2.1×105
ER466 1524 1524.0 825.4 7.2×1071524 1524.0 364.9 7.0×1071524 1524.0 28.5 9.5×105
ER941 5114 5177.4 1606.1 6.7×1075012 5013.3 1516.7 1.4×1085012 5014.1 458.5 1.2×107
ER2344 996411 1008876.4 1379.2 8.8×106953437 979729.2 1793.8 3.4×107902498 922339.5 2284.8 1.5×107
FF250 194 194.0 158.0 2.2×107194 194.0 1.9 4.6×105194 194.0 <0.1 2.3×103
FF500 263 265.0 197.5 9.4×106257 258.4 55.1 5.3×106257 257.0 0.4 8.7×103
FF1000 1262 1264.2 1743.0 3.8×1071260 1260.8 23.6 1.0×1061260 1260.0 84.9 2.6×105
FF2000 4548 4549.4 1571.0 2.4×1074546 4558.3 1160.8 2.8×1074545 4545.7 107.6 3.3×105
WS250 3415 3702.8 1424.4 7.6×1073085 3196.4 1983.5 4.2×1083083 3089.4 1140.5 6.1×107
WS500 2085 2098.7 1581.4 1.5×1082072 2083.3 1452.6 2.7×1082072 2082.6 179.3 2.4×106
WS1000 141759 161488.0 116.5 1.2×106123602 127493.4 2120.2 6.3×107109807 123682.6 2675.0 1.8×107
WS1500 13498 13902.5 1787.2 5.7×10713158 13255.7 1554.9 8.8×10713098 13255.1 1012.2 7.3×106
wins 12.5 13.0 * * 10.5 13.0 * * * * * *
Bovine 268 268.0 <0.1 3.0×102268 268.0 <0.1 2.4×103268 268.0 <0.1 6.8×101
Circuit 2099 2099.0 0.3 6.8×1042099 2099.0 1.2 5.9×1052099 2099.0 0.2 1.7×104
E.coli 806 806.0 <0.1 1.3×103806 806.0 <0.1 7.8×103806 806.0 <0.1 6.3×102
USAir97 4336 4336.0 254.9 1.1×1074336 4336.0 90.8 8.6×1064336 4336.0 756.5 2.5×106
HumanDi 1115 1115.0 5.8 4.8×1051115 1115.0 2.5 4.3×1051115 1115.0 0.6 1.7×104
TreniR 918 918.0 1.3 4.3×105918 918.0 2.4 1.4×106918 918.0 0.3 2.2×104
EU fli 348268 348347.0 914.8 1.6×106348268 348697.7 1495.0 6.0×106348268 351657.0 232.6 1.1×105
openfli 29300 29815.3 1835.0 7.4×10628834 29014.4 499.3 3.5×10626842 28704.3 2093.7 2.9×106
yeast 1413 1416.3 1461.9 1.1×1071412 1412.0 252.0 2.8×1061412 1412.0 21.7 1.0×105
H1000 314152 317805.7 1412.4 2.5×107314964 316814.8 1821.6 6.7×107306349 310626.5 2137.5 2.5×107
H2000 1275968 1292400.4 1200.0 8.6×1061275204 1285629.1 1620.1 2.5×1071243859 1263495.6 2861.9 1.2×107
H3000a 2911369 2927312.0 1598.5 7.4×1062885588 2906965.5 2041.5 1.8×1072844393 2884781.7 3280.7 1.1×107
H3000b 2907643 2927330.5 963.3 4.3×1062876585 2902893.9 1596.2 1.3×1072841270 2885087.0 3252.9 1.2×107
H3000c 2885836 2917685.8 1142.6 4.7×1062876026 2898879.3 1927.9 1.7×1072838429 2869348.5 3307.5 1.1×107
H3000d 2906121 2929569.2 1463.9 5.7×1062894492 2907485.4 2005.4 1.7×1072831311 2892562.7 3250.9 1.1×107
H3000e 2903845 2931806.8 1489.4 6.1×1062890861 2911409.3 1993.0 1.6×1072847909 2887525.7 3437.4 1.2×107
H4000 5194592 5233954.5 1749.1 5.3×1065167043 5190883.7 1954.2 1.2×1075044357 5137528.3 2907.0 6.6×106
H5000 8142430 8212165.9 1342.5 3.0×1068080473 8132896.2 2009.3 8.8×1067972525 8094812.6 3226.6 6.3×106
powergr 16158 16222.1 1532.2 5.6×10615982 16033.5 1610.3 9.8×10615862 15901.5 1286.4 1.8×106
Oclinks 611326 614858.5 990.9 2.5×106611344 616783.0 713.1 3.3×106612303 614544.0 584.6 4.5×105
faceboo 701073 742688.0 2234.4 4.8×106692799 765609.8 3132.9 1.5×107643162 739436.6 2978.5 2.2×106
grqc 15522 15715.7 2201.1 5.2×10613616 13634.8 2002.0 7.8×10613596 13629.2 871.8 9.2×105
hepth 130256 188753.7 2135.6 2.1×106108217 109889.5 2765.3 5.3×106106397 109655.6 3442.0 3.1×106
hepph 9771610 10377853.2 2286.8 7.2×1056392653 7055773.8 3120.6 2.9×1068628687 9370215.3 3376.3 1.4×106
astroph 59029312 60313225.8 3441.4 5.5×10555424575 57231348.7 3576.4 1.1×10662068966 62547898.1 1911.4 3.9×105
condmat 13420836 14823254.9 1481.3 2.9×1054086629 5806623.8 3511.9 1.6×1069454361 10061807.8 1779.5 4.9×105
wins 21.5 22.5 * * 19.0 19.5 * * * * * *
?Our FastCNP is an improved version of original FastCNP proposed in [42], which additionally integrating the node weighting scheme.
Note that, it is meaningless to calculate the wins in terms of average time and average node exchanges when different best objective values are achieved, and we represent them by *.
algorithms. Specifically, MACNP achieves the best objective
values for 23 out of the 26 real-world instances, including 18
new upper bounds and 5 known best objective values.
VI. AP PL IC ATIO N TO THE CARDINAL IT Y-CONSTRAINED
CRITICAL NODE PROBLEM
In this section, we show that our approach can also be used
to solve other critical node problems, by adapting MACNP to
the cardinality-constrained critical node problem (CC-CNP).
The experiments were again conducted on the synthetic and
real-world benchmarks described in Section V-A.
A. Cardinality-constrained critical node problem
CC-CNP is a cardinality constrained version of the classic
CNP [6]. CC-CNP aims to identify a minimum subset S⊆
Vsuch that any connected component in the residual graph
G[V\S]contains at most Wnodes where Wis a given
threshold value. To minimize the subset S, we solve a series
of CC-CNP with decreasing Kvalues. For a fixed K, we try
to find a set S⊆Vof Knodes, whose removal minimizes the
number of nodes in each connected component which exceeds
the cardinality threshold W[27]. For this purpose, we define
an auxiliary (minimization) function f0:
f0(S) =
T
X
i=1
max(|Ci| − W, 0) (2)
which calculates the total number of nodes in excess of Win
all Tconnected components of the residual graph. It is clear
that if f0(S)=0, then Sis a feasible solution of CC-CNP.
10
TABLE IV
COMPARISON BETWEEN MACNP AN D TH E STATE-O F-TH E-ART ALGORITHMS ON SYNTHETIC AND REAL-WORLD BENCHMARKS.
Instance K K BV Greedy3d Greedy4d Best VNS Best ILS GA CNA1FastCNPMACNP
BA500 50 195∗195 195 195 195 195 195 195 195
BA1000 75 558∗559 559 559 559 558 558 558 558
BA2500 100 3704∗3722 3722 3704 3722 3704 3704 3704 3704
BA5000 150 10196∗10196 10196 10196 10222 10196 10196 10196 10196
ER235 50 295∗315 313 295 313 295 295 295 295
ER466 80 1524 1938 1993 1542 1874 1560 1524 1524 1524
ER941 140 5012 8106 8419 5198 5544 5120 5114 5012 5012
ER2344 200 959500 1118785 1112685 997839 1038048 1039254 996411 953437 902498?
FF250 50 194∗199 197 194 195 194 194 194 194
FF500 110 257∗262 264 257 261 257 263 257 257
FF1000 150 1260∗1288 1271 1260 1276 1260 1262 1260 1260
FF2000 200 4545∗4647 4592 4549 4583 4546 4548 4546 4545?
WS250 70 3101 11694 11401 6610 3241 3240 3415 3085 3083?
WS500 125 2078 4818 11981 2130 2282 2199 2085 2072 2072?
WS1000 200 113638 316416 318003 139653 115914 113638 141759 123602 109807?
WS1500 265 13167 157621 243190 13792 14681 13662 13498 13158 13098?
wins 10.5 15.0 15.0 12.5 15.5 12.0 12.5 10.5 *
Bovine 3 268 268 268 268 268 268 268 268 268
Circuit 25 2099 2099 2100 2101 2117 2099 2099 2099 2099
E.coli 15 806 806 834 806 806 806 806 806 806
USAir97 33 4336 4442 4726 5444 4442 4336 4336 4336 4336
HumanDi 52 1115 1115 1115 1115 1115 1115 1115 1115 1115
TreniR 26 926 926 936 920 934 928 918 918 918?
EU fli 119 349927 349927 350757 356631 355798 351610 348268 348268 348268?
openfli 186 28834 29624 29552 31620 29416 28834 29300 28834 26842?
yeast1 202 1414 1416 1415 1421 1434 1414 1413 1412 1412?
H1000 100 328817 338574 336866 332286 344509 328817 314152 314964 306349?
H2000 200 1309063 1372109 1367779 1309063 1417341 1315198 1275968 1275204 1243859?
H3000a 300 3005183 3087215 3100938 3058656 3278172 3005183 2911369 2885588 2844393?
H3000b 300 2993393 3096420 3100748 3121639 3250497 2993393 2907643 2876585 2841270?
H3000c 300 2975213 3094459 3097451 3079570 3202002 2975213 2885836 2876026 2838429?
H3000d 300 2988605 3090753 3100216 3027839 3237495 2988605 2906121 2894492 2831311?
H3000e 300 3001078 3095793 3113514 3031975 3255390 3001078 2903845 2890861 2847909?
H4000 400 5403572 5534254 5530402 5498097 5877896 5403572 5194592 5167043 5044357?
H5000 500 8411789 8657681 8653358 8889904 9212984 8411789 8142430 8080473 7972525?
powergr 494 16099 16373 16406 16099 16533 16254 16158 15982 15862?
Oclinks 190 614504 614504 614546 623366 625671 620020 611326 611344 612303
faceboo 404 420334 608487 856642 865115 420334 561111 701073 692799 643162
grqc 524 13736 13787 13825 13751 13817 13736 15522 13616 13596?
hepth 988 114382 232021 326281 114933 123138 114382 130256 108217 106397?
hepph 1201 7336826 10305849 10162995 10989642 11759201 7336826 9771610 6392653 6156536?
astroph 1877 54517114 54713053 54517114 65937108 65822942 58045178 59029312 55424575 53963375?
condmat 2313 2298596 11771033 11758662 6121430 2298596 2612548 13420836 4086629 4871607
wins 22.5 24.0 25.0 24.5 22.5 22.5 21.5 21.0 *
∗Optimal results obtained by branch-and-cut algorithm [32] within 5 days.
?Improved best upper bounds.
The best results of CNA1 and FastCNP are obtained by running their source codes in our platform, which are slightly different their reported results in [27] and in [42], respectively.
Note that, for the results of our MACNP algorithm on hepph, astroph and condmat (in italic format), they are reached with tmax = 16,000 seconds.
B. Solving CC-CNP with MACNP
To solve CC-CNP, we adapt MACNP slightly and denote
the new algorithm by MACC-CNP. Basically, we replace the
objective function fused in MACNP with the minimization
function f0defined by Equation (2). To solve CC-CNP, we
start with an initial Kvalue (obtained with a construction
method, see below), and apply MACC-CNP to find a set Sof
Knodes whose removal minimizes the number of exceeded
nodes (i.e., minimizing f0(S)). If f0(S)=0, then Sis a
feasible solution of CC-CNP. At this moment, we decrease K
by one and solve the problem again. We repeat this process
until no feasible solution can be found and report the last
Sfound with f0(S) = 0. This general solution procedure is
inspired by a popular approach for solving the classic graph
coloring problem [21], [43], [45].
The initial Kvalue is obtained with an initial feasible solu-
tion S0. We first set S0to be empty. Then we iteratively pick
a node vfrom a large connected component whose cardinality
exceeds Wand move vto S0. We repeat this process until
a feasible solution S0is obtained (i.e., f0(S0)=0, meaning
that all components contain at most Wnodes). We set the
initial Kto equal |S0|.
C. Comparison with the state-of-the-art algorithms
Based on the synthetic and real-world benchmarks, we
compared our MACC-CNP algorithm with five state-of-the-
art algorithms: greedy algorithms (G1 and G2) [4], genetic
algorithm (GA) [4], multi-start greedy algorithm (CNA2)
[27], fast heuristic (FastCNP) [42]. Among these reference
algorithms, FastCNP was originally proposed for the classic
critical node problem, and we adapted it to CC-CNP in the
same way as for MACNP. The source code of CNA2 was
provided by its author [27]. For algorithms G1, G2 and GA,
whose source codes are not available, we used the results
reported in [4]. These results have been obtained with different
time limits tmax ∈[100,16000] seconds, which are, for most
instances, larger than our time limit of tmax = 3600 seconds.
For our comparative study, we ran CNA2 and FastCNP with
11
their default parameters under the time limit tmax = 3600
seconds, and each instance was solved 30 times. For our
MACC-CNP algorithm, we also solved each instance 30 times
independently under the same time limit.
The comparative results between our MACC-CNP algorithm
and the reference algorithms on the synthetic and real-world
benchmarks are displayed in Table V. To analyze these results,
we calculated the number of instances (wins) for which
MACC-CNP has a better result according to the two-tailed
sign test [12], as shown in the last row for each benchmark.
TABLE V
COMPARISON BETWEEN MACC-CNP AN D TH E STATE-O F-TH E-A RT
ALGORITHMS ON SYNTHETIC AND REAL-WORLD BENCHMARKS.
Instance L KB V G1 G2 GA CNA2 FA◦MA
BA500 4 47 47 47 47 47 47 47
BA1000 5 61 61 61 61 61 61 61
BA2500 10 100 101 100 100 100 100 100
BA5000 13 149 154 151 149 149 149 149
ER235 7 47 49 50 47 47 47 47
ER466 14 81 86 85 81 79 79 79?
ER941 25 139 149 152 139 141 135 135?
ER2344 1400 204 252 270 204 194 189 185?
FF250 5 48 48 49 48 48 48 48
FF500 4 100 102 102 100 100 100 100
FF1000 7 142 145 145 142 142 142 142
FF2000 12 182 191 187 182 182 182 182
WS250 40 73 79 80 72 71 70 70?
WS500 15 126 145 144 126 124 123 123?
WS1000 500 162 195 418 162 180 166 157?
WS1500 30 278 339 332 278 273 256 254?
wins 11.5 14.5 14.5 11.5 11.0 9.5 *
Bovine 15 4 4 4 4 4 4 4
Circuit 30 24 25 26 24 24 24 24
E.coli 20 15 16 15 15 15 15 15
USAir97 70 33 34 40 33 33 33 33
HumanDi 10 49 51 50 49 49 49 49
TreniR 10 28 30 31 28 27 27 27?
EU fli 850 113 127 118 113 113 112 112?
openfli 140 184 194 206 184 183 180 180?
yeast1 6 195 202 199 195 193 193 193?
H1000 800 103 172 151 103 97 95 92?
H2000 1600 221 362 313 221 207 195 188?
H3000a 2500 279 448 402 279 276 252 242?
H3000b 2500 279 456 401 279 270 251 244?
H3000c 2500 276 446 404 276 274 250 244?
H3000d 2500 276 452 402 276 272 250 244?
H3000e 2500 280 455 403 280 270 252 244?
H4000 3300 398 651 571 398 388 354 347?
H5000 4200 458 745 662 458 459 413 410?
powergr 20 428 449 440 428 430 397 397?
Oclinks 1100 197 209 200 197 193 193 192?
faceboo 450 324 472 821 324 523 375 378
grqc 20 480 497 501 480 486 462 461?
hepth 70 981 1040 1042 981 1029 955 944?
hepph 3600 1228 1416 1572 1228 1103 994 1120
astroph 12000 1322 3284 1769 1322 1364 1249 1329
condmat 500 2506 2506 2651 2506 2357 2357 2320?
wins 21.5 25.5 25.0 21.5 22.5 19.0 *
?Improved best upper bounds.
◦Adaptation of the FastCNP algorithm [42] to CC-CNP (denoted as FA).
The proposed MACC-CNP (MA) algorithm.
Table V shows that MACC-CNP achieves the best objective
values for all synthetic instances, and yielding in particular 7
new upper bounds. At a significance level of 0.05, MACC-
CNP is significantly better than G1 and G2. Compared to GA,
CNA2 and FastCNP, MACNP is better but the differences are
not significant, winning 11.5, 11.0 and 9.5 instances. MACC-
CNP is also very effective on the real-world benchmark. At a
significance level 0.05, MACC-CNP significantly outperforms
all reference algorithms. MACC-CNP discovers new upper
bounds for 18 instances and reaches the best upper bounds
for 5 out of 8 remaining instances. These observations show
that MACC-CNP is highly competitive for solving CC-CNP.
VII. DISCUSSION AND ANALYSIS
We now study some key ingredients of the proposed algo-
rithm. This study was performed on 4 representative synthetic
instances from different families (BA5000, ER941, FF500,
WS250) and 4 representative real-world instances (TreniR,
H3000a, H4000, hepth). We ran each algorithm variant 15
times on each instance with a time limit tmax = 3600 seconds.
A. Benefit of the two-phase node exchange strategy
To study the benefit of the large component-based two-
phase node exchange strategy (Section IV-D), we compare
MACNP with an alternative version MACNP0where the
neighborhood search using the large component-based two-
phase node exchange is replaced by using the conventional two
node exchange strategy. In other words, at each neighborhood
search iteration of MACNP0, we choose at random a node
u∈S(Sbeing the current solution) and a node v∈V\Sand
accept the node exchange if the exchange leads to a solution
better than S; Otherwise, we test another random pair (u, v).
MACNP0and MACNP share thus the same ingredients except
the neighborhood search procedure.
TABLE VI
COMPARISON BETWEEN LARGE COMPONENT-BAS ED TW O-PHASE NODE
EXCHANGE (MACNP) AND TRADITIONAL NODE EXCHANGE (MACNP0).
MACNP0MACNP
Instance fbest favg #exch
sec fbest favg #exch
sec
BA5000 10198 10200.8 1.5 10196 10196.0 5708.9
ER941 5014 5171.1 5.8 5012 5013.7 29428.5
FF500 257 257.0 134.9 257 257.0 21445.3
WS250 3083 3257.7 37.6 3083 3101.1 59954.5
TreniR 918 918.0 1322.5 918 918.0 102275.0
H3000a 3451908 3483709.9 <0.12849170 2883554.5 3909.4
H4000 6165357 6224768.6 <0.15081209 5144354.2 2117.6
hepth 23964174 24528095.0 <0.1106552 108354.0 1100.8
The results for MACNP and MACNP0are summarized in
Table VI. For each instance, we report the best objective
value (fbest), the average objective value (favg ), and the
average number of node exchanges per second (#exch
sec ) over
15 trials achieved by each algorithm. The results show that
MACNP performs significantly better than MACNP0in terms
of all comparison indicators primarily due to its much lower
computational complexity to perform a node exchange. We
observe that in each second, MACNP performs seventy or even
tens of thousands times more node exchanges than MACNP0.
Finally, even if we do not show a direct comparison between
our component-based neighborhood with the two neighbor-
hoods proposed in [3], Table IV (columns 10 and 7) shows that
FastCNP (using our component-based neighborhood) domi-
nates the best ILS algorithm of [3] (using two complementary
refined neighborhoods), which further demonstrates the inter-
est of the component-based neighborhood.
12
B. Effectiveness of the node weighting scheme
To assess the interest of the node weighting scheme in
the Component-Based Neighborhood Search (CBNS) (Section
IV-D3), we compared MACNP with its alternative algorithm
MACNP1where the node weighting scheme is disabled in
CBNS. In MACNP1, we also decompose the exchanging
operation into two phases (“add-phase” and “removal-phase”),
and execute them separately. However, for “add-phase”, a
node is randomly removed from a large connected component
instead of selecting the node by the node weighting scheme.
Table VII shows the comparison of MACNP and MACNP1
on the tested instances, based on four indicators: best objective
value (fbest), average objective value (favg), average time to
find the best objective value (tavg), and average number of
node exchanges required to attain the best objective value
(#exch). An obvious observation is that the algorithm with the
node weighting scheme (i.e., MACNP) significantly outper-
forms MACNP1(which lacks the node weighting scheme) on
almost all instances except WS250. For WS250, both MACNP
and MACNP1achieve the best objective value 3083, while the
average objective value 3093.8 of MACNP1is slightly better
than 3101.1 of MACNP. More importantly, MACNP needs less
time and fewer node exchanges to achieve the best objective
values. These observations demonstrate the effectiveness of
the node weighting scheme.
C. Benefit of the double backbone-based crossover
To study the benefit of the double backbone-based crossover
(Section IV-E), we compare MACNP with an alternative
version MACNP2. MACNP2is obtained from MACNP by
replacing the double backbone-based crossover with a single
backbone-based crossover which only treats the common ele-
ments as the backbone. The single backbone-based crossover
operator first constructs a partial solution S0by inheriting
all the common elements of two parent solutions and then
completes the partial solution S0by adding nodes from a large
component of the residual graph until |S0|=K.
Comparative results of MACNP and MACNP2in terms
of the best objective value and average objective value are
displayed in the left and right part of Figure 5 respectively.
The X-axis indicates the instance, and the Y-axis shows the
gap of our results (eight best values or average values) to
the known best values in percentage, which is defined as
(f−KBV )×100/KBV where fis the best or average
objective value, and KBV is the known best objective (see
3rd column of Table IV). A negative gap indicates an improved
upper bound for the corresponding instance.
Figure 5 shows that compared to MACNP2, MACNP is able
to attain a superior best objective value for all 8 instances
including 4 new upper bounds (see values below 0 on the left
part of Figure 5). MACNP also outperforms MACNP2on all
8 tested instances in terms of the average objective value, as
shown in the right part of Figure 5. This experiment confirms
the value of our double backbone-based crossover.
VIII. CONCLUSIONS AND FUTURE WORK
In this work, we proposed an effective memetic search ap-
proach for solving the classic critical node problem (MACNP),
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
BA5000 ER941 FF500 WS250 TreniR H3000a H4000 hepth
GAP to KBV (%)
Instance
Best Value of MACNP
Best Value of MACNP2
Known Best Value
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
BA5000 ER941 FF500 WS250 TreniR H3000a H4000 hepth
GAP to KBV (%)
Instance
Average Value of MACNP
Average Value of MACNP2
Known Best Value
Fig. 5. Comparative results of MACNP (with double backbone-based
crossover) and MACNP2(with traditional backbone-based crossover).
which combines a component-based neighborhood search for
local optimization, a double backbone-based crossover opera-
tor for solution recombination and a rank-based pool updating
strategy to guarantee a healthy diversity of the population. To
ensure its effectiveness, the component-based neighborhood
search relies on a focused and reduced neighborhood owing to
its two-phase node exchange strategy and the node weighting
scheme. Additionally, the double backbone-based crossover
not only conserves solution features from the parent solutions,
but also introduces diversity by including exclusive elements
from parent solutions in a probabilistic way.
To demonstrate the competitiveness of the proposed algo-
rithm, we evaluated MACNP on a broad range of synthetic and
real-world benchmarks. The computational results showed that
MACNP significantly outperforms state-of-the-art algorithms
on both benchmarks. We also assessed the performance of
MACNP for solving the cardinality-constrained critical node
problem (MACC-CNP), which is an important variant of the
classic CNP. Our results showed that the approach is also
highly competitive compared with state-of-the-art algorithms.
Finally, we performed experiments to investigate the benefit
of different search components and techniques.
Future work motivated by our findings will be to investigate
opportunities for further improving the performance of our al-
gorithm by incorporating machine learning techniques. Given
that most ideas of this work are not limited to CNP, another
inviting avenue for research is to adapt the proposed method
to solve other critical node problems with different measures
(e.g., distance-based connectivity and betweenness centrality).
ACKNOWLEDGMENT
We are grateful to our anonymous reviewers for their in-
sightful comments that have helped us to significantly improve
this work. We thank Dr. W. Pullan for kindly sharing the
source codes of the CNA1 and CNA2 algorithms.
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TABLE VII
COMPARISON OF MACNP ( WITH NODE WEIGHTING)AGA IN ST MACNP1(WITHOUT NODE WEIGHING).
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APPENDIX
To highlight the contributions of the key algorithmic com-
ponents of the MACNP algorithm, we compare MACNP
with four variants (named MACNP0, MACNP1, MACNP2,
MACNP3) where a particular component of MACNP is dis-
abled or replaced by a traditional alternative operator.
•MACNP0is obtained from MACNP by replacing the two-
phase node exchange strategy of Section IV-D2 with the
traditional node exchange strategy (e.g., [3]). See Section
VII-A for more details.
•MACNP1is obtained from MACNP by disabling the
node weighting technique of Section IV-D3. See Section
VII-B for more details.
•MACNP2is obtained from MACNP by replacing the
double backbone-based crossover of Section IV-E with
a traditional backbone-based crossover (e.g., [40], [44]).
See Section VII-C for more details.
•MACNP3is obtained from MACNP by replacing the
local optimization component (i.e., the component-based
neighborhood search of Section IV-D) with a classic
mutation operator, which works as follows. Each time an
offspring solution is created by the crossover, ngswaps
are randomly performed on the offspring solution. The
value of ng(1≤ng≤ |S|) is given by ng=d1.0/xe
where |S|is the number of the selected node in the
current solution and xis a random value in (0,1]. If
ng>|S|, we set ng=|S|.
Table VIII summarizes the comparative results between
MACNP and these variants on the set of 8 representative
instances used in Section VII. These results are based on 15
runs of each algorithm to solve each instance with a time
limit tmax = 3600 seconds per run. These results show that
removing any component of the MACNP algorithm deterio-
rates its performance and the MACNP algorithm performs the
best thanks to the symbiosis of its ingredients.
15
TABLE VIII
COMPARATIVE RESULTS BETWEEN MACNP AND ITS FOUR VARIANTS. TH E BE ST RE SULT S ARE I ND ICAT ED IN B OL D.
MACNP MACNP0MACNP1MACNP2MACNP3
Instance fbest favg fbest favg fbest favg fbest fav g fbest favg
BA5000 10196.0 10196.0 10198.0 10200.8 10196.0 10196.0 10196.0 10196.0 10196.0 10209.1
ER941 5012.0 5013.7 5014.0 5171.1 5014.0 5014.4 5012.0 5026.1 5019.0 5163.1
FF500 257.0 257.0 257.0 257.0 257.0 257.0 257.0 257.0 257.0 257.5
WS250 3083.0 3101.1 3083.0 3257.7 3083.0 3093.8 3373.0 3520.9 3085.0 3838.9
TreniR 918.0 918.0 918.0 918.0 918.0 918.0 918.0 918.0 918.0 918.0
H3000a 2849170.0 2883554.5 3451908.0 3483709.9 2862217.0 2913716.6 3072097.0 3102585.7 2931805.0 2989920.8
H4000 5081209.0 5144354.2 6165357.0 6224768.6 5187236.0 5313477.9 5531484.0 5569792.9 5322446.0 5377694.8
hepth 106552.0 108354.0 23964174.0 24528095.0 108453.0 111804.3 121295.0 126286.5 124871.0 137912.0
avg. 1007049.6 1019468.6 4200113.6 4282047.3 1022171.8 1044809.8 1093079.0 1102322.9 1049824.6 1065739.3
