CHN and Swap Heuristic to Solve the Maximum Independent Set Problem

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 6, December 2017, pp. 3583 – 3592
ISSN: 2088-8708 3583
Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
CHN and Swap Heuristic to Solve the Maximum
Independent Set Problem
Bouhouch Adil1
, Loqman Chakir2
, and El Qadi Abderrahime3
1
Team TIM, High School of Technology, CEDoc-SFA faculty of sciences, Moulay Ismail University, Meknes, Morocco
2
department of informatics, Sciences Faculty, Dhar Mehraz, Sidi Mohammed Ben Abdellah University,Fez, Morocco
3
LASTIMI, High School of Technology - Mohammed V University of Rabat, Morocco
Article Info
Article history:
Received: Jan 9, 2017
Revised: Jun 14, 2017
Accepted: Jul 3, 2017
Keyword:
Max-Stable problem
Independent set
CHN
Local search
Combinatory problems
Graph
ABSTRACT
We describe a new approach to solve the problem to find the maximum independent
set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper,
we show how Max-Stable problem can be reformulated into a linear problem under
quadratic constraints, and then we resolve the QP result by a hybrid approach based
Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the
solution given by the CHN will be the starting point of the local search. The new
approach showed a good performance than the original one which executes a suite of
CHN runs, at each execution a new leaner constraint is added into the resolved model.
To prove the efficiency of our approach, we present some computational experiments
of solving random generated problem and typical MSSP instances of real life problem.
Copyright c 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Name Bouhouch Adil
Affiliation Team TIM, High School of Technology Meknes
Address High School of Technology Meknes Moulay Ismail university Meknes Morocco
Phone +212664786422
Email bouhouch.adil@gmail.com
1. INTRODUCTION
The Max-Stable Set Problem (MSSP) is a problem that attempts to find the largest independent set
at a given graph. All the nodes included in the independent set must respect the condition that they are not
pairwise connected by an arc. Max-table is largely applied in many areas: case-based reasoning [1], com-
puter vision [2], scheduling, ... . Max-Stableis a Strong NP-Hard problem while it’s hard to be approximate.
Therefore, solving MSSP in polynomial time for arbitrary graph case is unlikely. For arbitrary graph there
are many exact algorithm which enumerating all cliques and select the one with the maximal cardinality. To
our knowledge Harary [3] was the first one who introduced in the literature an exact method. Loukakis [4]
generated all maximal independent sets lexicographically introduced by a depth-first enumerative algorithm.
Their study includes a comparison against Regneri [5] and Tsukiyama [6] algorithmes. The theoretical superior
efficiency of their algorithm is also reinforced by computational results, moreover there method was largely
faster than that proposed by Tsukiyama [6] and that introduced by Bron [7]. After two years, Loukakis [8]
imported additional change to their previous work [4] and improve it by three speed-up. In the years 1988s,
Johnson [9] introduced an exact approach which determines all maximal stable sets in lexicographic order. The
approach get each independent set by times complexity of order O(n3
). Chiba [10] proposed an algorithm to
the maximal cliques on the order O(a(G)mµ) of times complexity -where a(G) is the arboricity of graph G- ,
this is over the time complexity of [6]. Based on the Born [7] work, Tomita [11] proposed an improved variant
having complexity equal to O(3n/3
). Intuitively, it seems that exacts approaches are better to solve Max-stable
problem. They consist to enumerate all possible independent set and then select the one which have the maxi-
mum of cardinality [12]. However, the analysis of the complexity discouraged this idea for large graph. In this
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Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
, DOI: 10.11591/ijece.v7i6.pp3583-3592

3584 ISSN: 2088-8708
case heuristic methods proved to be an efficient alternative away to solve large problem instances. The second
categories contains Meta-heuristic methods which explore the search space, local search methods and hybrid
methods. In [13] show how to build a good hybrid strategies. Recently, many approach based neural network
was developed to solve combinatorial and hard optimisation problem. The authors of [14]investigate Continu-
ous Hopfield Neural Network (CHN) to solve largess instances of Max-Stable problems. The mean idea is to
execute the CHN many times. The role of first run is to find a valid initial solution of MSSP by CHN and a
quadratic reformulation. In the second step the cardinality of the solution given by first run is added as linear
constraint to the resolution model, then they run CHN again. The second step can be repeated until to have no
solution improvement. In this paper we propose new hybrid approach, in order to take advantage of he faster
convergence of CHN in the one hand. In the second hand, the Local Search which look through neighborhods
to find the best one. To solve MSSP by LS, There are many successful heuristic [15, 16, 17, 18, 19, 20] . The
common among them is The start with a random solution and improve it regularly by very simple operations
like deletions of nodes which don’t meet the adjacent condition , insertions of new nodes or swaps (case when
current node succeed by its neighbors).
This paper is structured as follows: In section 2. we present the resolution model bested CHN which
is divided into many steps:we introduce the continuous Hopfeild network, next we give the reformulation of
maximum stable set problem as a 0-1 quadratic program, then we build an adapted energy function of CHN.
Section 3. is devoted to improve the solution by LS. Experimental results are presented in the last section.
2. CONTINUOUS HOPFEILD NEURAL NETWORK TO SOLVE MSSP
In this section we present an overview of [14] approach which implement CHN to solve a quadratic
model of MSSP.
2.1. The continuous Hopfield neural network
CHN is a fully connected neural Model with one layer. It was Introduced by Hopfield in the year 1980
to solve combinatorial problems. Further, Hopfield [21] proposed an energy functions to solve many opti-
mization problems as linear programming problems, analog to digital conversion, graph coloring problem,TSP,
processing image and . The evolution of CHN dynamic is controlled by The following differential equation:
dy
dt
= −
x
τ
+ T x + ib
(1)
where
x : vector of neurons input
y : vector of output
T : the Matrix of weight between each neurones pairs
Neurons output is governed by function:
xi = g(yi) =
1
2
(1 + tanh(
yi
u0
)) with u0 > 0 and i = 1, ..., n
The g function is bounded (g(u) ∈ [0, 1]) and u0 is a parameter to estimate the gain (or slope) of
the activation function. The limit point of the CHN ue
by this differential system exists such that u(t) = ue
∀t ≥ te (and te ≥ 0), this point is called an equilibrium point The energy of CHN is a Lyapunov function
defined as:
E(x) = −
1
2
xt
Tx − (ib
)t
x +
n
i=1
1
τi
xi
0
g−1
(v)dv
If T is symmetric then the equilibrium points existence is guaranteed. So, any combinatorial problems formu-
late as the following expression can be solved by The CHN:
E(x) = −
1
2
xt
Tx − (ib
)t
x (2)
IJECE Vol. 7, No. 6, December 2017: 3583 – 3592

IJECE ISSN: 2088-8708 3585
The matrix T is called also matrix of weights connections and just inhibitory connection is allowed on this
symmetric matrix. E(x) evaluate at the hypercube [0, 1]n
and converge to the corners of this n-dimensional
space. To solve a combinatorial problem by CHN we need just to associate the energy function of CHN with the
objective function of problem to be minimized, so that the minimum of E(x) coincide with the combinatorial
problem solution. Implicitly, the inputs of network outputs represent the problem solution. To show how we
can mapping a combinatorial problem to be associated with it neural model we take the assignment problem.
It is a easier and direct model to be mapped. So, we consider the following model of assignment problem with
n variables m linear constraint:
(QP)



Min 1
2 xt
Qx + qt
x
Subjectto
Ax = b
xi ∈ {0, 1} i = 1, .....n
Furthermore, we he need to define the following set to solve this QP:
• H ≡ {x ∈ [0, 1]n
}: the Hamming hypercube
• Hc ≡ {x ∈ {0, 1}n
}: the corners of the Hamming hypercube.
• Hf ≡ {x ∈ Hc : Ax = b}: feasible solution set.
To map the QP bellow, we must respect some conditions so that local minimums of the QP to be
associated with the CHN limit. The energy can also be defined by two terms:
E(x) = E0
(x) + ER
(x) ∀x ∈ H
Where:
• the term E0
(x) is associated with the problem objective function.
• the quadratic function ER
(x) have two objectives. The first one, is to penalizes the connections which
violated at last one constraint. The second one, concerns to guarantees that the CHN converge to a valid
solution. To perform a good mapping, this function must be constant in FxH and ding a well choice.
For this problem an adapted generalized function is proposed in [22]:
E(x) =
α
2
xt
Qx +
1
2
(Ax)t
φ(Ax) + xt
diag(γ)(1 − x) + βt
Ax ∀x ∈ H (3)
With parameters α ∈ R+
, γ ∈ Rn
, β ∈ RN
and a m × m matrix φ. The goal of this work is to solve the
maximum cardinality of the independent set by improving the previous approach based on the CHN proposed
by [14]. The main idea of last work is to convert the MSSP as CHN energy function. Before mapping problem
used a quadratic 0-1 model to represent the MSSP. In the section 2.2., we describe the reformulation of MSSP,
then we present the solving approach in the section 2.3..
2.2. Formulation of the Maximum Stable Set Problem
Let G = (V, E) an undirected graph with V a set of n nodes and E the set of m edges. An independent
set of a graph G is a set of nodes S with the property that any node nodes in S is not connected by a direct
edges to others nodes of S. The MSSP consist to determine the independent set which have the cardinality
maximum α(G).
So, the objective function to maximize is the number of nodes which are pairwise independent, rather we re-
formulate the connection penalty by a quadratic constraint which represent direct edge connection. To perform
the reformulation, we define the binary variables xi such that:
xi =
1 if vi ∈ S
0 Otherwise
Let S ⊂ V be a stable set of nodes. For each node vi of the graph G, we have the following relations:
CHN and Swap Heuristic to Solve the Maximum Independent Set ... (Bouhouch adil)

3586 ISSN: 2088-8708
• To be valid S must don’t contains two adjacent node. So, for two node vi and vj in S: there corresponding
neurones xi and xj we have: let pij a connection penalty between vi and vj.
(vi, vj) ∈ S =⇒ xi = 1andxj = 1
(vi, vj) ∈ E =⇒ pijxixj = 0 =⇒ pij = 0
(vi, vj) /∈ E =⇒ pijxixj = 1 =⇒ pij = 1
Now we can define the quantity:
P(x) =
n
i=1
n
j=1
pijxixj (4)
Referring to all connections constraints imposed in 2.2.. We have P(h) = 0 when all nodes in S are
pairwise not adjacent. With
pij =
1 if (vi, vj) ∈ E
0 Otherwise
(5)
• The objective function to be minimised is :
f(x) = −
n
i=1
xi
Finally, the QP of the MSSP problem can be formulated as:
(QP)



Min f(x) = −
n
i=1
xi
Subject to
P(x) =
n
i=1
n
j=1
pijxixj = 0
x ∈ {0, 1}n
After the reformulation of MSSP problem as a quadratic 0-1 programming, it can be solved by any
adapted approach by minimizing the linear function under quadratic constraints (QP), such as interior point,
semidefinite relaxations [23] or lagrangian relaxations[24]. In this paper we are interested in a very different
approach based on CHN [14]. In the last approach authors propose to solve the MSSP into two phases. First,
solving the QP of MSSP by CHN. We note by Γ be the value found by CHN. Second, changing QP model
formulation by adding a new constraint to the objective function such as:
(NQP)



Min f(x) = −
n
i=1
xi
Subject to
P(x) =
n
i=1
n
j=1
pijxixj = 0
n
i=1
xi ≥ Γ
x ∈ {0, 1}n
Then they solve the new quadratic problem(NQP) by a second CHN associated to this new reformulation
(CHN2
) . In this work we propose to replace the second run CHN2
by the local search heuristic. On other
words, the solution given by the first run on the QP will be the starting point of the local search.

IJECE ISSN: 2088-8708 3587
2.3. A continuous Hopfield network to solve MSSP
Now The MSSP was reformulate as QP, the next step is to find an adapted energy function , we choose
the same one introduced by [14]:
E(x) = −α
n
i=1
xi +
1
2
φ
n
i=1
n
j=1
bijxixj + γ
n
i=1
xi(1 − xi) (6)
The advantage of this energy expression is more associated with the objective function, it’s relaxed by the
quadratic constraint.
by corresponding the Equation (2) with Equation (6), we deduce the weights and thresholds:
Ti,j = −φpij + 2δi,jγ
ib
i = α − γ
(7)
With δij represent the Kronecker symbol.
δij =
1 if i = j
0 if i = j
Matrix p is given by Equation (5), the values of parameters φ, γ and α are critical to reach the CHN
equilibrium points into a valid solution of MSSP.
The study of partial derivatives of the generalized energy function led to control those parameters to
guide the convergence to a feasible solution:
∂E(x)
∂xi
= Ei(x) = −α + φ
n
j=1
bijxj + γ(1 − 2xi)
To determine the parameters-setting, Talavan [25] used the hyperplane method analyse to study ∂E(x).
This procedure consist of partition the Hamming hypercube H by a hyperplane containing all feasible
solutions so that two properties must be respected by the CHN evolution: first, adding any solution not belong-
ing to this hyperplane, second, dropping out any infeasible solution belonging to the hyperplane.
Also, some conditions are imposed to determine these parameters-setting easily:
φ > 0, γ ≥ 0
• To minimize the objective function, we fixed the following constraint:
α > 0
• To escape the stability of the interior points x ∈ H − HC, the next constraint is necessary:
Ti,i = 2γ ≥ 0
While MSSP model contains just one constraint then we have:
HC − HF = {x ∈ HC/h(x) > 0}
Let x ∈ HC −HF , so for two adjacents nodes xi and xj are in the stable set S, then xi = xj = 1 and therefore
the activation of xi will be discouraged if E0
i (x) ≥ where > 0.
Based to this study the parameter settings are restricted by the following condition:
−α + φ − γ ≥
A feasible solution can be reached if the following conditions are respected:
α > 0, φ > 0, γ ≥ 0
−α + φ − γ =
(8)
The question then is about making good choice of these parameter under the condition bellow to calculate the
weights and thresholds of the CHN.,The advantage of resolving MSSP by Continuous dynamic of Hopfield
neural network is that CHN are very fast but always converge to a local minima, in [14] authors insert some
linear constraint to perturb the network weights and consequently escaping from all local minima less than
threshold Γ. This method appear efficient, but the control of CHN convergence become difficult. .

3588 ISSN: 2088-8708
3. PROPOSED APPROACH
Hopfeild neural network converge faster, but it converges closely to the nearest local minima of the
starting point. So many time the network gives a no good solution. To overcome this weakness many solution
was proposed to escape from local minima. In [14], authors propose to add a linear constraint which restrict a
lower bound of the minimum cardinality of the solution, as shown in [14], the insertion of new constraint to the
QP leads to improve the solution, but in contrast of obtained numerical results, we remark that network failed
to give a valid solution many time. To improve this meta-heuristic approach, we propose to use a hybridization
with other heuristic. It seems beneficial to combine CHN with an adapted local search. In [16] authors prove
that swap(1,2) give a good performance. Algorithm (1) show the proposed local search based Swap heuristic.
As described, this process by replacing each node n in the the given initial solution by two nodes u and v. The
chosen u and v must be neighbors of n. Therefore, the solution is improved at each time by adding one node.
This last search is down in linear times. This is possible by adding a collection of neighborhods set V (n) which
store for each graph node the set of it adjacent nodes. The Local search work directly on the solution given
by the CHN. We note that to check if a node n of the graph is in the solution, we verify if its associated CHN
node is active. Also, to reduce the time to find the neighbors of the candidate node which can be replaced,
we maintaining a stack To visit of valid candidate to visit. After examine each node, we removed it from the
candidate list, and go to the next until the list To visit become empty. Although, we added to the candidates
list To visit any new insert node in the solution S.
In this work, we implement the direct and simple local search based on swap heuristic. For this, to replace
the current candidate node n, we consider just the first pair of node u, v founded, which meet conditions.
Nevertheless, we can add more improvement to this LS, like sorting V(n) by number adjacent neighbor in the
solution [26] or plateau search [16].
Function Swap 1 2( S,T : ) : Solution
In :
• T o visit : Stack of candidate nodes
• V (i) : Set of neighborhood of the nodei
Out : Improved Solution
T o visit = S
While (T o visit is not empty) do
n = pop(T o visit)
For (u, v) ∈ V (n) × V (n) do
Have Adjacent in S = false
If (u /∈ S and u /∈ S and T (u, v) = −φ ) then
If (is marked(u) = false and is marked(v) = false) then
For (a = n) ∈ S do
If (T (u, a) = −φ or T (v, a) = −φ ) then
Have Adjacent in S = true
Break
end If
end For
If (Have Adjacent in S = false ) then
replace n by u and v
add u and v T o visit
Break
end If
end If
end If
end For
mark(n)
done
End
Algorithm 1. Local search algorithm
Naturally, CHN attempted to minimize the number conflicted constraints, therefore, some times network can
stabilise with not null Energy E = 0, and gives an invalid solution. To overcome this problem, we add a local
heuristic called Remove Process. It delete nodes from the solution until to being valid.

IJECE ISSN: 2088-8708 3589
Sort S by number of Adjacent
For each v ∈ S do
if(∃u ∈ S / (u, v) ∈ E) delate v
end For
Algorithm 2. Delate Process
4. NUMERICAL RESULT
To show the efficiency of our approach, we present a computational results. First, we run the solver
on the random indirected generated graphs. The used generator return a Gn,p random graph, also known as an
Erdõs-Rényi graph or a binomial graph [27], where n the number of nodes and the graph density which can
be determinate by the probability for edge creation p. So we generated five classes classified by the number
of graph nodes from 100 to 500, and for each class, the density p varied from 10% to 100%. For a given p
we generate random 100 instances. Figure 1 plot the CPU time tacked by CHN, CHN2
-the improved variant
[14]- and CHNSwap. The comparison is down under fixing the density D = 50% and varying the numbers of
nodes. Practically, the curves of CHN and CHNSwap are superposed.
Figure 1. the CPU time as a function of number of nodes.
Figure 2 give the evolution of times of CHNSwap approach as function of the graph density (D),
respectively, for classes V=100, V=300 and V=500. We note that left vertical axis is the times performed by
CHN and the right for LS. It is clear that the problem difficulty increases as the number of nodes increases. The
same remark can be seen at all tested graph classes.
To study the solution quality we run our approach on selected instance from the benchmark DIMACS[28].

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Figure 2. the CPU times tacked by CHN and LS in CHNSwap (a)Class of graph containing 100 nodes (b)Class
of graph containing 300 nodes (c)Class of graph containing 500 nodes
CHN CHNSwap
Graphs V E α(G) Min Mean Best Min Mean Best
brock200 2.clq 200 9876 12 8 8 8 9 9 9
brock200 4.clq 200 13089 17 11 11 11 13 13,99 14
C250.9.clq 250 27984 24 34 34 34 37 37,69 38
c-fat200-1.clq 200 1534 58 12 12 12 12 12 12
c-fat200-2.clq 200 3235 44 23 24 24 24 24 24
c-fat200-5.clq 200 8473 30 58 58 58 58 58 58
DSJC125.9.col 125 6961 44 30 30,08 31 32 32 32
frb30-15-1.clq 450 83198 55 20 20 20 24 24 24
gen200 p0.9 44.clq 200 17910 44 31 31 31 35 35,77 36
hamming6-4.clq 64 704 4 2 2,42 3 4 4 4
johnson16-2-4.clq 120 5460 8 6 6,35 7 8 8 8
johnson8-2-4.clq 28 210 14 2 4 6 4 4 4
johnson8-4-4.clq 70 1855 11 8 8,91 9 10 13,27 14
keller4.clq 171 9435 126 11 6 5 7 7 7
MANN a27.clq 378 70551 126 117 117 117 117 117,98 121
MANN a9.clq 45 918 16 12 12 12 15 15,76 16
p hat300-1.clq 300 10933 8 6 6 6 6 6,51 8
p hat300-2.clq 300 21928 25 22 22 22 24 24,99 25
p hat300-3.clq 300 33390 36 29 30,89 31 33 33,97 34
san200 0.7 1.clq 200 13930 18 15 15 15 15 15 16
san200 0.9 3.clq 200 17910 44 31 31 31 33 33 33
san400 0.9 1.clq 400 71820 100 39 44,15 46 52 52,98 53
Table 1. Comparison of solution quality between CHN and CHNSwap over DIMACS

IJECE ISSN: 2088-8708 3591
Graphs E V CHN(ms) Swap(ms)
johnson8-2-4.clq 210 28 2,12 0,01
MANN a9.clq 918 45 3,18 0,06
hamming6-4.clq 704 64 1,62 0,18
johnson8-4-4.clq 1855 70 9,53 0,12
johnson16-2-4.clq 5460 120 25,65 0,32
DSJC125.9.col 6961 125 7,14 0,32
C125.9.clq 6963 125 9,88 0,08
C125.9.clq 6963 125 9,44 0,06
keller4.clq 9435 171 41,06 0,31
c-fat200-1.clq 1534 200 77,86 1,17
c-fat200-2.clq 3235 200 61,01 0,95
c-fat200-5.clq 8473 200 34,64 0,61
brock200 2.clq 9876 200 76,04 0,72
brock200 4.clq 13089 200 65,54 0,4
san200 0.7 1.clq 13930 200 16,09 0,5
gen200 p0.9 44.clq 17910 200 4,58 0,44
san200 0.9 3.clq 17910 200 2,5 0,68
C250.9.clq 27984 250 8,92 0,71
p hat300-1.clq 10933 300 163,3 2,38
p hat300-2.clq 21928 300 159,36 2,43
p hat300-3.clq 33390 300 139,28 1,07
MANN a27.clq 70551 378 160,81 0,82
san400 0.9 1.clq 71820 400 115,3 1,82
frb30-15-1.clq 83198 450 362,35 2,31
Table 2. Times performance
The efficiency of the proposed approach is reinforced by the experimental results. Table 1 shows that
our method gives good result then used CHN alone. Table 2 show the time added by the improvement process
with local heuristics against the time needed by CHN. It’s clear that improvement is down with adding non-
significant time.
Our approach is very powerful from a theoretical point of view. This approach gives a good performance
compared with the improved approach CHN2
proposed by [14] which crush often with no valid solution.
5. CONCLUSION
In the last decade several areas have applied MSSP problem and Max-Clique problem. So, many
heuristic were developed. In this sense we directed our contribution. In this work, a new approach has been
proposed based CHN and swap local heuristic to solve the maximal cardinality of independent sets. The MSSP
is solved into three step: First, it is reformulate as quadratic problem. Second, the QP is associated with CHN
energy function, then we stabilise CHN to converge at the solution. Third, the local search process performed
by starting from the solution given in the second step. It same difficult to integrating the local heuristic in the
main loop of CHN stabilisation for this we have chosen the collaborative hybridisation between CHN and LS.
We executed a series of experiments in order to prove the efficiency regarding the time complexity and the
solution of quality. The time allowed to LS is very small so that we can assert that our approach is able to find
the maximal independent set for very large graph. Furthermore, the rate of success to give a valid solution is
up to 100% in our approach. We note that this approach can be used to solve the Max-Clique problem special
for perfect graph by solving the graph dual.
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3592 ISSN: 2088-8708
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