Bees Swarm Optimization Metaheuristic Guided by Decomposition for

Solving MAX-SAT

Youcef Djenouri

, Zineb Habbas

and Wassila Aggoune-Mtalaa

LRDSI, Computer Science Department, Saad Dahleb University, Blida, Algeria

Department of Computer Science, University of Lorraine, Metz, France

LIST, Luxembourg Institute of Science and Technology G.D., Esch-sur-Alzette, Luxembourg

Keywords:

BSO, MAX-SAT, Decomposition Methods, Kmeans, BSOGD.

Abstract:

Decomposition methods aim to split a problem into a collection a collection of smaller interconnected sub-

problems. Several research works have explored decomposition methods for solving large optimization prob-

lems. Due to its theroretical properties, Tree decomposition has been especially the subject of numerous

successfull studies in the context of exact optimization solvers. More recently, Tree decomposition has been

successfully used to guide the Variable Neighbor Search (VNS) local search method. Our present contribution

follows this last direction and proposes two approaches called BSOGD1 and BSOGD2 for guiding the Bees

Swarm Optimization (BSO) metaheuristic by using a decomposition method. More pragmatically, this paper

deals with the MAX-SAT problem and uses the Kmeans algorithm as a decomposition method. Several ex-

perimental results conducted on DIMACS benchmarks and some other hard SAT instances lead to promising

results in terms of the quality of the solutions. Moreover, these experiments highlight a good stability of the

two approaches, more especially, when dealing with hard instances like the Parity8 family from DIMACS.

Beyond these ﬁrst promising results, note that this approach can be easily applied to many other optimization

problems such as the Weighted MAX-SAT, the MAX-CSP or the coloring problem and can be used with other

decomposition methods as well as other metaheuristics.

1 INTRODUCTION

The NP-Complete satisﬁability problem (SAT) is of

central importance in computation theory. SAT for-

malism is used to model many academic or real prob-

lems like coloring problem, decision support and

automated reasoning. Formally, SAT is deﬁned as

follows: given a set of n boolean variables V =

, v

, ..., v

}, a Conjunctive Normal Form (CNF) is

a conjunction of clauses, each clause being a disjunc-

tion of literals, while a literal is a variable v

from V

or its negation, noted ¬v

. A clause is satisﬁed when

at least one of its literals is set to true. A CNF is

satisﬁed if an assignment of some variables in V sat-

isﬁes all the clauses. The SAT problem asks for an as-

signment of some variables in V that satisﬁes a CNF

F. The problem is said SAT if such an assignment

exists and UNSAT otherwise. This paper addresses

the NP-Hard MAX-SAT problem, a generalization of

SAT. Given a CNF formula F, the objective of MAX-

SAT is to satisfy as many clauses of F as possible.

A solution of a MAX-SAT instance is a complete in-

stantiation of variables in V that satisﬁes a maximal

number of clauses. As with many NP-complete prob-

lems, existing algorithms dedicated to SAT are either

complete or incomplete. A complete algorithm aims

to solve the problem while an incomplete algorithm

aims only to ﬁnd satisfying instantiations. The most

effective complete algorithms are based on the DPLL

procedure (Davis et al., 1960). They mainly differ

by the heuristics used for the branching rule (Dubois

et al., 1996)5. MAX-SAT is a generalization of SAT.

But while SAT is a decision problem, MAX-SAT is its

optimization version. Of course, ﬁnding the optimal

solution for a NP-Complete optimization problems is

too time expensive because of the exponential time

complexity.

To deal with this problem, many incomplete meth-

ods are proposed for solving MAX-SAT problems.

GSAT (Li, 1997) is a randomized local search. It

starts drawing randomly a valuation for the variables

and then makes a certain number of ﬂips on variables

that reduce the number of unsatisﬁable clauses. This

process is repeated until getting the optimal solution

472

Djenouri, Y., Habbas, Z. and Aggoune-Mtalaa, W.

Bees Swarm Optimization Metaheuristic Guided by Decomposition for Solving MAX-SAT.

DOI: 10.5220/0005810004720479

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 472-479

ISBN: 978-989-758-172-4

or reaching a limit on the number of attempts. Walk-

sat (Selman et al., 1994) is an extended version of

GSAT. A noise represented by a probabilistic instruc-

tion, is introduced in the procedure to achieve the

random walk move. A variable drawn randomly is

considered with a probability p and with (1− p) the

variable that yields to the maximum satisﬁed clauses

is selected. Then it improves the obtained solution

by a local search method. In (Drias et al., 2005), a

Bees Swarm Optimization metaheuristic (BSO) has

been proposed for solving Weighted Maximum Sat-

isﬁability Problem. Currently, a trend for improv-

ing incomplete algorithms, consists in combining

wisely the best properties from different approaches.

In (Lardeau et al., 2006), a new hybrid algorithm

(called GASAT) embedded a tabu search procedure

into the evolutionary framework. The GASAT perfor-

mance comes from its original and highly specialized

crossover operators, a powerful tabu search method

and the interaction between these two methods. To

enhance the performance of MAX-SAT solvers and

in order to deal with hard and large instances, some

solving approaches propose to explore the structural

properties of the problem.

This paper follows this line of research and its

main purpose is to improve the BSO metaheuristic

by decomposing the problem before solving it. Ti

decompose a problem several clustering techniques

can be found in the literature. In the context of data

mining, unsupervised learning algorithms correspond

to the most popular class. In general, the cluster-

ing consists in grouping together or putting in the

same cluster homogeneous data. In this paper, the

Kmeans algorithm was selected for the decomposi-

tion step. Based on structural knowledges coming

from Kmeans, this paper proposes two extended BSO

algorithms guided by decomposition named BSOGD1

and BSOGD2. In BSOGD1 each bee of the colony

considers as its region only a part of the problem that

coincides with a particular cluster. The bee returns

a partial modiﬁcation of the “reference solution” be-

cause it can access to variables in only one cluster. A

bee in BSOGD2 can access to all the clusters and has

consequently a more important knowledge about the

structural properties of the instance to be solved. To

validate the proposed approaches, experimental stud-

ies have been carried out on DIMACS and some hard

Uniform− Random−3− SAT instances, the ﬁrst re-

sults reveal that the our approaches outperform the

state of the art MAX-SAT approaches.

The reminder of this paper is organized as fol-

lows. Section 2 presents the BSO-MAXSAT algo-

rithm. Section 3 concerns our main contributions and

mainly presents the two proposed algorithms. The ex-

perimental results of our proposition are reported in

section 4. Finally, section 5 concludes the present pa-

per by some remarks and perspectives.

2 THE BSO-MAXSAT

ALGORITHM

In (Drias et al., 2005), a Bees Swarm Optimization al-

gorithm for solving the Weighted MAX-SAT problem

was proposed. MAX-SAT is a particular Weighted

MAX-SAT problem in which the weight associated

with each clause is 1. The main principle of this ap-

proach called BSO-MAXSAT is formally resumed by

algorithm 1.

Algorithm 1: BSO-MAXSAT algorithm.

Input: A MAX-SAT instance P

Begin

1: Sref ← Initial

Solution

2: while non stop do

3: TabuList ← Sref

4: FindSearchRegion (Sref, k, S

, S

, . . . , S

)

5: for each bee i do

6: LocalSearch ( S

, BestSol

)

7: TableDance ← BestSol

8: end for

9: Sref ← BestSolution(TableDance)

10: end while

End

First, the initial bee BeeInit creates the solu-

tion reference named Sref and saves it in a Tabu

list. From this solution Sref, a set of k regions R =

, S

, . . . , S

} is determined thanks to the proce-

dure FindSearchRegion. After that, each bee b

assigned to a region S

in order to explore it using

the local search procedure (LocalSearch). Finally,

the communication between bees is performed via the

TableDance, in order to elect the best solution that

will be the solution reference for the next iteration.

• The Evaluation of the Solution: A solution s of

BSO-MAXSAT is an instantiation of n variables,

where the i

element is set to 0 if the variable

is assigned to false and set to 1 if the variable is

assigned to true. The evaluation of s is based on

the number of clauses satisﬁed by S.

• Determination of Regions: Theaim of procedure

FindSearchRegion is to divide the space of solu-

tions into k disjoint regions. Given the solution

reference Sref, a parameter Flip is introduced in

order to ensure the diversiﬁcation step. Then, k

Bees Swarm Optimization Metaheuristic Guided by Decomposition for Solving MAX-SAT

473

disjoint solutions are generated where the i

so-

lution is obtained by changing successfully from

Sref the bits: {(1 × Flip) + i, (2× Flip) + i, (3×

Flip) + i, ...n− i}.

• Local Search Process: The aim of procedure Lo-

calSearch is to explore a region by identifying in

each step the neighbors of a given solution. Given

the solution s, this operation ensures the intensiﬁ-

cation by changing only one bit of s at a time.

This algorithm was tested on the well known BMC

instances

. The obtained results were very promis-

ing by ﬁnding the optimal solution in most of the

cases. However, the performance of BSO-MAXSAT

decreases when dealing with large instances and hard

ones. To cope with this problem, two approaches are

proposed in the next section to guide BSO-MAXSAT

by exploring some structural knowledge coming from

a decomposition.

3 BSO GUIDED BY

DECOMPOSITION

This section presents the main contributions of this

paper. First the principle of Kmeans for SAT problem

is described in subsection 3.1. Then two different ap-

proaches to guide BSO by using decomposition will

be formally described in subsection 3.2 and subsec-

tion 3.3.

3.1 Kmeans for Decomposing SAT

K-means is one of the simplest unsupervised learning

algorithms which can solve the well known clustering

problem. The procedure follows a simple and easy

way to classify a given data set through a certain num-

ber of clusters (assume k clusters) ﬁxed a priori. The

main idea is to deﬁne k centroids, one for each clus-

ter. The centroids should be placed in a cunning way

because the clustering result depends on their loca-

tion in the clusters. In order to optimize the efﬁciency

of the outcomes, it is judicious to place them as far

as possible from each other. The next step is to take

each point belonging to a given data set and associate

it to the nearest centroid. When no point is pending,

the ﬁrst step is completed and an early grouping is

done. At this stage we need to re-calculate k new cen-

troids for the new clusters resulting from the previous

step and iterate the process. The latter stops when no

more changes of the clusters are observed, in other

words when no centroid move any more. To adapt the

http://www.cs.ubc.ca/

hoos/SATLIB/benchm.html

Kmeans procedure on clauses clustering, we propose

a new similarity and center of gravity computation for

clauses.

3.1.1 Similarity Between Clauses

The similarity between two clauses represents the

degree of consistency between them. On the con-

trary their dissimilarity denotes the inconsistency be-

tween them. Intuitively they are similar when they

share a lot of variables and they are dissimilar if they

are different. Let us propose as a similarity mea-

sure denoted Dist

clauses between two clauses c

and

the following formula: Dist clauses(c

, c

) = n −

ncv(c

, c

) where ncv(c

, c

) is the number of com-

mon variables between c

and c

This distance is a validmetric because it meets the fol-

lowing mathematical properties of a metric distance

function, which are:

• ∀(c

, c

) ∈ C

Dist

clauses(c

, c

) ∈ R

• ∀c ∈ C Dist

clauses(c, c) = 0

• ∀(c

, c

) ∈ C

Dist

clauses(c

, c

) =

Dist

clauses(c

, c

)

• ∀(c

, c

) ∈ C

Dist

clauses(c

, c

) ≤

Dist clauses(c

, c

) + Dist clauses(c

, c

)

Example 1. Consider a SAT instance deﬁned as the

set of variables V = {v

, v

} and the two fol-

lowing clauses c

and c

• c

: v

, v

• c

: v

, v

The common variables of c

and c

are {v

, v

} so

ncv(c

, c

) = 2 and Dist clauses(c

, c

) = 1

3.1.2 Centroid Computation

Consider the set of clauses C = {c

, c

, ..., c

}. The

aim is to ﬁnd the clause corresponding to the centroid.

The idea is to compute the frequency of each variable

among all the clauses in a same cluster. The length

of the clause center noted l which corresponds to the

average number of items of all the r clauses is deter-

mined as follows: l =

∑

i=1

. Then, the variables of

the r clauses are sorted according to their concurrency

in the r clauses and only the l frequent variables are

kept in a vector called Freq as follows:



center

clause[Freq[ j]] = 1

∀ i 6= Freq[ j] center clause[i] = 0

Example 2. Let be the MAX-SAT following problem:

F = (¬v

) ∧ (¬v

∨ v

) ∧ (¬v

∨ ¬v

) ∧ (v

∨

) ∧(¬v

∨ v

) ∧(¬v

∨ v

)

Note that this problem is UNSAT because there is no

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

474

possible value for v

or v

that satisﬁes the clause v

∨

. The success ratio of this instance is 83% because

only 5 clauses out of 6 can be satisﬁed.

Table 1: Variables and their Frequency.

Variables Frequency

To compute the center-clause of the six ones c

to c

, we ﬁrst compute the centroid l. Here l =

1+2+3+2+2+2

= 2. Then, the l frequent variables are

selected. According to the table 1, {v

, v

} are fre-

quent variables. So center

clause = {v

, v

3.2 BSOGD1: A First BSO Algorithm

Guided by Decomposition

This section aims to present a ﬁrst BSO algorithm

Guided by Decomposition called BSOGD1. Intu-

itively, BSOGD1 proceeds like BSO-MAXSAT, ex-

cept that each bee explores its region by taking into

account the structural knowledge coming from the de-

composition step (the clusters and/or separators). The

principle of BSOGD1 is formally described by algo-

rithm 2.

Algorithm 2: Algorithm BSOGD1.

Input: A MAX-SAT instance P

Begin

1: Decompose(P, k, G

, G

, ..., G

)

2: Sref ← Initial

Solution

3: FindSearchRegion1 ( k,G

, G

, ... , G

, R

..., R

)

4: while non stop do

5: TabuList ← Sref

6: for each bee i do

7: LocalSearch1( Sref , R

, BestSol

)

8: TableDance ← BestSol

9: end for

10: Sref ← BestSolution1(TableDance)

11: end while

12: End

BSOGD1 consists of two main steps described as

follows:

• The ﬁrst step (Procedure decompose) partitions

the constraint network corresponding to the ini-

tial problem P to be solved in order to identify

some relevant structural components such as the

clusters and the separators for instance. The de-

composition method used in this algorithm is the

Kmeans one presented in subsection 3.1. The pro-

cedure Decompose(P, k, G

, G

, ..., G

) splits a

given MAX-SAT instance P into a collection of k

clusters G

, G

, ..., G

, where k is the number of

bees. Each cluster is a subset of the initial set of

clauses of P. A clause belongs to only one clus-

ter. Two clusters Gi and G j are connected if they

share at least one variable v, which is a variable

used both by a clause of Gi and Gj.

• The second step concerns the principle of BSO

for solving MAX-SAT. The speciﬁcity of this ap-

proach is to use the procedure FindSearchRe-

gion1 instead of the conventional procedure of

BSO for determining the regions. The proce-

dure FindSearchRegion1 takes as input a set

of clusters G

, G

, ..., G

obtained from the

Kmeans procedure and returns a collection of re-

gions R

, R

. . . , R

. Initially, a set of variables

Var(G

) in G

is assigned to each region R

. When

a variable v belongs simultaneously to both R

and

it is removed from the cluster that contains the

minimum clauses including v. This heuristic gen-

erates independent regions allowing each bee to

improve locally the solution. The procedure Lo-

calSearch1 improvesin the solution Sref only the

variables of its region and returns in TableDance a

solution with a partial improvement. The function

BestSolution1 determines the next solution Sref

as follows:

Sref ← s ← BestSol

] ⊙ . . . ⊙ BestSol

]

where BestSol

] is the partial solution found by

a bee i and ⊙ is a simple concatenation of the par-

tial solutions.

3.3 BSOGD2: A Second BSO

Algorithm Guided by

Decomposition

BSOGD2 differs from BSOGD1 by the determination

of regions, the local search procedure and the election

of the next solution Sref . BSOGD2 determines the

regions in a conventional way, like BSO-MAXSAT.

Howeverunlike BSO-MAXSAT, each bee explores its

own region by considering in its search all the clusters

, G

, ..., G

. This heuristic enables a bee to guide

more ﬁnely its research because ih has a global in-

formation unlike than the bee in BSOGD1. The next

solution Sref is the best solution among all the solu-

tions found by the bees. The principle of BSOGD2 is

formally described by algorithm 3.

Bees Swarm Optimization Metaheuristic Guided by Decomposition for Solving MAX-SAT

475

Algorithm 3: Algorithm GDBSO2.

Input: A MAX-SAT instance P

Begin

1: Decompose(P, k, G

, G

, ..., G

)

2: Sref ← Initial

Solution

3: while non stop do

4: TabuList ← Sref

5: FindSearchRegion2(Sref, k, S

, . ..,

)

6: for each bee i do

7: LocalSearch2(S

, G

, ..., G

, Sol

)

8: TableDance ← Sol

9: end for

10: Sref ← BestSolution(TableDance)

11: end while

12: End

4 PERFORMANCE ANALYSIS

4.1 Experimental Conditions

To validate the proposed approaches several experi-

ments were carried out in a Single Machine Pentium-

I3 with 4Go memory. The proposed approaches have

been implemented using Java environment and tested

on 21 instances of the well known DIMACS instances

and 10 Uniform Random-3-SAT instances. All the

considered instances are available at SATLIB site

The DIMACS instances used in this study were di-

vided into three classes aim-50, aim-100, and par-

ity8. The aim-50 class contains 8 different instances

involving 50 variables and a number of clauses which

varies from 80 to 100. The aim-100 class includes 8

instances deﬁned on 100 variables, while the number

of clauses varies from 160 to 200. The last class con-

tains 5 instances of Parity8 with 350 variables and a

number of clauses varying from 1149 to 1171. More-

over, for the Uniform Random-3-SAT instances, the

number of variables varies from 20 to 250 while the

number of clauses varies from 91 to 1065.

In this section, the results of the following tests are

presented:

1. First, the results obtained with the BSO-MAXSAT

algorithm on the three classes of DIMACS in-

stances are compared to the ones obtained with

the GASAT algorithm (the well-known incom-

plete algorithm for the MAXSAT problem).

2. Then, the two proposed algorithms BSOGD1 and

BSOGD2 are compared to the BSO-MAXSAT

http://www.cs.ubc.ca/

hoos/SATLIB/benchm.html

algorithm on the same instances (the DIMACS

ones).

3. Finally, in order to further analyze the gain of our

contribution, the two algorithms BSOGD1 and

BSOGD2 are confronted to each other on the Uni-

form Random-3-SAT instances.

To validate our approach from a practical point of

view, the following performance measures are consid-

ered: the CPU time, the average success rate (SR%)

and the best success rate (Best

SR%). Note that the

results reported in all the tables are an average of the

results obtained on 100 executions. Moreover, the

column (n,m) gives the size of an instance where n

is the number of variables and m is the number of

clauses. The execution times were not reported since

all the times are comparable, although the slight dif-

ferences are in favor of the BSO algorithm.

4.2 Comparison of the GASAT and the

Hybrid BSO-MAXSAT Algorithms

As mentioned in section 1, in GASAT, a simple local

search is added at the end of the Genetic Algorithm

to improve the quality of the ﬁnal solution. In or-

der to compare BSO-MAXSAT to GASAT, the same

local search used in GASAT is added to the classical

BSO-MAXSAT. The resulting algorithm is called Hy-

brid Bees Swarm Optimization for Maximal Satisﬁa-

bility Problem (HBSO-MAXSAT).

Table 2 summarizes the results of the comparison

between these two algorithms. Note that for GASAT,

only the best success rates are presented by the au-

thors, while for HBSO-MAXSAT, both the average

success rate and the best success rates are presented.

Clearly, our approach presents a better stability. In-

deed the average results are almost closed to the best

ones, and are often around the order of 92% for the

ﬁrst two groups of instances. Furthermore, the results

are also comparable for all instances (around 90%) for

each instance. For the third family, the best success

rates are of about 60%, because this class includes

very hard instances. However the stability of this ap-

proach is still kept. Unlike HBSO-MAXSAT, GASAT

presents too distant success rates from one instance

to another (only 6% for the instance 1-6yes1-2 and

100% for the instance 1-6-yes1-3). The results are

very bad for the third family of problems. Finally,

regarding the average results, HBSO-MAXSAT out-

performs everywhere GASAT. This encouraging re-

sult conﬁrms the relevance of our idea that aims to

guide the BSO algorithm by decomposition.

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

476

Table 2: Comparing HBSO-MAXSAT and GASAT algorithms.

Class Instances (n,m) HBSO-MAXSAT GASAT

SR (%) Best SR (%) Best SR (%)

1-6yes1-1 (50,80) 92,39 97,75 100

1-6yes1-2 (50,80) 93,12 97,5 6

1-6yes1-3 (50,80) 90,27 93,75 100

aim50 1-6yes1-4 (50,80) 90,01 93,75 100

2-0yes1-1 (50,100) 91,96 95 68

2-0yes1-2 (50,100) 90,53 95 100

2-0yes1-3 (50,100) 92,13 97 100

2-0yes1-4 (50,100) 90,39 95 100

Average 91,35 95,59 84,25

1-6yes1-1 (100,160) 90,93 95 2

1-6yes1-2 (100,160) 91,96 96,25 0

1-6yes1-3 (100,160) 92,91 96,25 0

aim100 1-6yes1-4 (100,160) 92,37 96,25 0

2-0yes1-1 (100,200) 90,65 94 10

2-0yes1-2 (100,200) 90,34 93,5 74

2-0yes1-3 (100,200) 91,33 94 98

2-0yes1-4 (100,200) 92,06 95 18

Average 91,74 95,25 25,25

1 (350,1149) 55,51 60,57 17,02

2 (350,1157) 56,23 61,8 25,53

Parity8 3 (350,1171) 55 59,78 21,28

4 (350,1155) 54,18 58,78 15,22

5 (350,1171) 54,86 59,95 21,74

Average 55,16 60,18 20,16

4.3 Performance of the Approaches

Guided by Decomposition

The aim of this second series of tests is to show the

beneﬁt obtained with the two proposed approaches.

For this purpose, the Hybrid version of BSOGD1

(constructed in the same manner as the HBSO-

MAXSAT algorithm) called (HBSOGD1 and the Hy-

brid BSOGD2 called (HBSOGD2 are compared to

HBSO-MAXSAT.

4.3.1 Results Obtained on the DIMACS

Instances

Table 3 shows the success rates and the best success

rates obtained using HBSO-MAXSAT, HBSOGD1 and

HBSOGD2 on the DIMACS instances. According

to this table, one can remark that thanks to the ex-

ploitation of the structural knowledge extracted from

the Kmeans decomposition (the clusters are intercon-

nected via small separators), both HBSOGD1 and

HBSOGD2 behave better than the HBSO-MAXSAT

approach in all the cases. Moreover, when deal-

ing with hard instances of DIMACS such as the Par-

ity8, the success rate grows from 60% for HBSO-

MAXSAT to 72% for HBSOGD1 and 73% for HB-

SOGD2. These results are very promising.

4.3.2 Results Obtained on the Uniform

Random-3-SAT Instances

This last series of tests aims to more analyse the

behaviour of the proposed approaches when dealing

with harder MAXSAT instances. Therefore, the HB-

SOGD1 and HBSOGD2 algorithms are compared on

the Uniform Random-3-SAT instances. Table 4 shows

the success rates and the best success rates of HBSO-

MAXSAT, HBSOGD1 and HBSOGD2 for different in-

stances.

This table reveals that HBSOGD2 improves HB-

SOGD1. Indeed, the success rate of the second algo-

rithm is up to 90% in all the used instances, while it

does not exceed 90% in some cases for HBSOGD1.

This is because the bees in the second approach have

a global vision of the decomposed problem and each

bee can access to all clusters and their separators if

necessary. In the second approach, each bee im-

proves the worst cluster, which is the one that has

the least satisﬁed clauses. Nevertheless, the bees in

the ﬁrst approach have access to only a restricted part

of the problem, which more speciﬁcally refers to a

cluster. Moreover, the obtained results show that the

two approaches improve again the results of HBSO-

MAXSAT by more than 13% in most cases.

Bees Swarm Optimization Metaheuristic Guided by Decomposition for Solving MAX-SAT

477

Table 3: Positionning HBSOGD1 and HBSOGD2 versus HBSO-MAXSAT: results obtained on the DIMACS instances.

Class Instances (n,m) HBSO-MAXSAT HBSOGD1 HBSOGD2

SR (%) Best SR (%) SR (%) Best SR (%) SR (%) Best SR (%)

1-6yes1-1 (50,80) 92,39 97,75 96,06 98,75 96,01 98,75

1-6yes1-2 (50,80) 93,12 97,5 96,14, 98,75 96,4 98,75

1-6yes1-3 (50,80) 90,27 93,75 94,31 97,5 93,99 96,25

aim50 1-6yes1-4 (50,80) 90,01 93,75 93,77 96,25 93,89 96,25

2-0yes1-1 (50,100) 91,96 95 96,07 98 96,06 98

2-0yes1-2 (50,100) 90,53 95 94,53 97 94,17 97

2-0yes1-3 (50,100) 92,13 97 95,86 98 96,05 99

2-0yes1-4 (50,100) 90,39 95 94,53 97 94,51 97

Average 91,35 95,59 95,16 97,66 95,13 97,62

1-6yes1-1 (100,160) 90,93 95 94,53 97 94,51 97

1-6yes1-2 (100,160) 91,96 96,25 95,66 97,5 95,76 98,12

1-6yes1-3 (100,160) 92,91 96,25 96,39 98,75 96,49 98,12

aim100 1-6yes1-4 (100,160) 92,37 96,25 96,48 98,75 96,41 98,75

2-0yes1-1 (100,200) 90,65 94 95,08 97 95,21 96,5

2-0yes1-2 (100,200) 90,34 93,5 94,64 96,5 94,65 96,5

2-0yes1-3 (100,200) 91,33 94 94,5 97,5 96,93 98

2-0yes1-4 (100,200) 92,06 95 95,91 97,5 95,99 98

Average 91,74 95,25 95,67 97,56 95,80 97,62

1 (350,1149) 55,51 60,57 72,55 74,32 72,52 74,32

2 (350,1157) 56,23 61,8 73,2 74,93 73,22 74,5

Parity8 3 (350,1171) 55 59,78 72,16 74,04 72,16 73,7

4 (350,1155) 54,18 58,78 71,26 72,81 71,26 72,81

5 (350,1171) 54,86 59,95 72,04 73,53 72 74,29

Average 55,16 60,18 72,42 73,93 72,23 73,92

Table 4: Positionning HBSOGD1 and HBSOGD2 ver-

sus HBSO-MAXSAT: results obtained on the Uniform

Random-3-SAT instances.

Instances HBSO-MAXSAT HBSOGD1 HBSOGD2

uf20-91 80,79 88,94 92,13

uf50-218 79,35 90,33 93,52

uf75-325 81,13 90,84 94,22

uf100-430 80,40 90,43 90,43

uf125-538 79,64 89,42 92,65

uf150-645 80,68 89,51 90,25

uf175-763 80,89 89,47 90,31

uf200-860 80,57 89,72 92,32

uf225-960 80,33 89,54 90,24

uf250-1065 80,48 89,47 93,26

5 CONCLUSION

In this paper, two extended Bees Swarm Optimiza-

tion algorithms guided by decomposition namely

BSOGD1 and BSOGD2 where proposed for address-

ing the maximal satisﬁability problem. The Kmeans

procedure has been chosen for the decomposition

step. In BSOGD1, each bee of the colony consid-

ers as its region only a part of the problem, which

corresponds to a particular cluster. The bee returns

a partial modiﬁcation of the reference solution be-

cause it can access the variables in only one cluster.

A bee in BSOGD2 can access to all the clusters. To

demonstrate the performance of the two approaches,

two main series of experimentation have been carried

out. First, the results on the DIMACS instances indi-

cate that the two approaches outperform the classical

BSO algorithm. Then, the results obtained on the hard

instances of Uniform-Random-3-SAT reveal that the

second approach beneﬁts from the best exploration of

the decomposition and improves the results obtained

by the ﬁrst approach. As a short term perspective, we

plan to investigate other metaheuristics to analyze in a

deeper way the effect of a decomposition on the max-

imal satisﬁability problem. We also plan to apply the

two proposed approaches to other optimization prob-

lems like the Weighted MAXSAT, the coloring Prob-

lem, and Constraint Satisfaction Problems.

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