HYPERTREE DECOMPOSITION FOR SOLVING CONSTRAINT

SATISFACTION PROBLEMS

Abdelmalek Ait-Amokhtar, Kamal Amroun

University of Bejaia, Algeria

Zineb Habbas

LITA, University of Metz, France

Keywords:

CSP, Structural decompositions, Hypertree decomposition, Acyclic Solving algorithm, Enumerative search.

Abstract:

This paper deals with the structural decomposition methods and their use for solving Constraint Satisfaction

problems (CSP). mong the numerous structural decomposition methods, hypertree decomposition has been

shown to be the most general CSP decomposition. However so far the exact methods are not able to ﬁnd

optimal decomposition of realistic instances in a reasonable CPU time. We present Alea, a new heuristic to

compute hypertree decomposition. Some experiments on a serial of benchmarks coming from the literature or

the industry permit us to observe that Alea is in general better or comparable to BE (Bucket Elimination), the

best well known heuristic, while it generally outperforms DBE (Dual Bucket Elimination), another successful

heuristic. We also experiment an algorithm (acyclic solving algorithm) for solving an acyclic CSP obtained

by using the heuristic Alea. The experimental results we obtain are promising comparing to those obtained by

solving CSP using an enumerative search algorithm.

1 INTRODUCTION

Many important real world problems can be formu-

lated as Constraint Satisfaction Problems (CSP)s. A

CSP consists of a set V of variables each with a do-

main D of possible values and a set C of constraints

on the allowed values for speciﬁed subsets of vari-

ables. A solution to CSP is the assignment of values

to variables which satisﬁes all the constraints. The

usual method for solving CSPs is based on backtrack-

ing search. This approach has an exponential theo-

retical time complexity in O(e.d

) for an instance of

CSP for which e is the number of constraints, n is

the number of variables and d is the cardinality of

the largest domain of variables. Even if CSPs are

NP-complete in general, some decomposition meth-

ods have been successfully used to characterize some

tractable classes (Dechter and Pearl, 1989; Gyssens

et al., 1994; Jeavons et al., 1994; Gottlob et al., 2000),

biconnected components (Freuder, 1982), hinge de-

composition (Gyssens et al., 1994), hinge decompo-

sition combined with tree clustering (Gyssens et al.,

1994), hypertree decomposition and generalized hy-

pertree decompositions (Gottlob et al., 1999a; Gott-

lob et al., 2005) and more recently Spread Cut De-

composition (Cohen et al., 2005). The main principle

of these methods is to decompose the CSP into sub-

problems that are organized in a tree or a hypertree

structure. The sub-problems can be solved indepen-

dently in order to solve the initial CSP. All these meth-

ods are characterized by their computational com-

plexity and the width or hypertree-width of respec-

tively the tree or the hypertree they generate. Among

these numerous methods, Gottlob and al, (Gottlob

et al., 1999a), have shown that hypertree decomposi-

tion dominates all the other structural decomposition

methods excepted the recently introduced spread-cut

decomposition method (Cohen et al., 2005). The ﬁrst

approach proposed to compute the hypertree decom-

position is the exact algorithm (opt-k-decomp (Got-

tlob et al., 1999b)). For any parameter k, opt-k-

decomp determines if a hypertree decomposition of

width w bounded by k exists. If the response is yes,

the algorithm computes this decomposition . Unfor-

tunately, opt-k-decomp and all its improvements are

not efﬁcient in practice and run out of time and space

for realistic instances. To overcome this limitation,

some heuristics have been explored to ﬁnd a hyper-

Ait-Amokhtar A., Amroun K. and Habbas Z. (2009).

HYPERTREE DECOMPOSITION FOR SOLVING CONSTRAINT SATISFACTION PROBLEMS.

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence, pages 85-92

DOI: 10.5220/0001662400850092

 SciTePress

tree decomposition in a reasonable time, even if it is

not optimal( (Musliu and Schafhauser, 2007), (Ko-

rimort, 2003), (McMahan, 2003)). In this paper we

consider a new heuristic to compute a hypertree with-

out using the tree decomposition. To do this, we ﬁrst

consider a particular deﬁnition of hypertree decompo-

sition and then we propose a generic algorithm which

proceeds in two steps. In the ﬁrst step, we look for

a ﬁrst separator of the hypergraph which will con-

stitute the root of the output hypertree. A separator

is a set of hyperedges whose removal either makes

the problem smaller or decomposes a given problem

into at least two components each of which is smaller

than the orignal one. In the second step, this separa-

tor is used for partitioning the hypergraph by remov-

ing its vertices. Afterwards, each component is re-

cursively partitioned, according to the connectedness

condition to be satisﬁed by the resulting decomposi-

tion. This heuristic can be considerably inﬂuenced

by the choice of this ﬁrst separator. In order to com-

pute the ﬁrst separator we propose the heuristic Alea

which considers the ﬁrst separator as the ﬁrst con-

straint introduced by the user or the ﬁrst constraint

in the ﬁle describing a given CSP instance. Some

experiments on a serial of benchmarks coming from

the literature or the industry (Ganzow et al., 2005)

permit us to observe that Alea is in general better

or comparable to BE(Bucket Elimination), the best

well known heuristic as far as we know, while it gen-

erally outperforms DBE (Dual Bucket Elimination),

another successful heuristic. We also experiment the

resolution of constraint satisfaction problem based on

a hypertree decomposition and the results we obtain

are promising comparing to those obtained by solv-

ing CSP using nFC2-MRV, a forward-checking algo-

rithm with a Minimum Remaing Value ordering algo-

rithm (C.Bessi`ere and Larossa, 2002).

2 PRELIMINARIES

The notion of Constraint Satisfaction Problems (CSP)

was introduced by Montanari in (Montanari, 1974).

Deﬁnition 1 (Constraint Satisfaction Problem). A

constraint satisfaction problem is deﬁned as a 3-uplet

P =< V,D,C > where X = {x

,...,x

} is a set of

n variables, D = {d

,...,d

} is a set of ﬁnite do-

mains; each variable x

takes its values in its domain

. C = {c

,...,c

} is a set of m constraints. Each

constraint c

is a pair (S(c

),R(c

)) where S(c

) ⊂ X,

is a subset of variables, called scope of c

and R(c

) ⊂

∏

∈S(c

)

is the constraintrelation, that speciﬁes the

allowed combinations of values.

In order to study the structural properties of con-

straints satisfaction problems, we need to present the

following deﬁnitions relative to the graphical repre-

sentation of CSPs.

Deﬁnition 2 (Graph (Dechter, 2003)). A graph

G =< V,E > is a structure that consists of a ﬁnite

set of vertices V and a set of edges E. Each edge is

incident to an unordered pair of vertices {u,v}.

Deﬁnition 3 (Hypergraph (Dechter, 2003)). A hy-

pergraph is a structure H =< V, E > that consists of

a set of vertices V and a set of hyperedges E . Mainly

a hyperedge h ∈ E is a subset of vertices V. The hy-

peredges differ from regular edges in that they may

connect more than two variables.

The structure of a CSP P =< V,D,C > is captured

by a hypergraph H =< V,E > where the set of ver-

tices V is the same as the set of variables V and the

set of hyperedges E corresponds to the set of con-

straints C. For the set of hyperedges K ⊆ E let the

term vars(K) =

e∈K

e i.e the set of the variables oc-

curing in the edges K. For a set L ⊆ V let the term

edgevars(L) = vars({e|e ∈ E,e∩ L 6=

0}) i.e all the

variables occuring in a set of edges where each edge

in the set contains at least one variable in L.

Deﬁnition 4 (Hypertree (Gottlob et al., 2001)). Let

H =< V,E > a given hypergraph. A hypertree for hy-

pergraph H is a triple < T, χ,λ > where T = (N,E) is

a rooted tree and χ and λ are two functions labelling

each node of T. The functions χ and λ map each node

p ∈ N by two sets χ(p) ⊆ V and λ(p) ⊆ E

A tree is a pair < T,χ > where T = (N,E) is a rooted

tree and χ is a labelled function as previously deﬁned.

Deﬁnition 5 (Generalized Hypertree Decomposi-

tion (Gottlob et al., 2001)). A generalized hypertree

decomposition of a hypergraph H =< V, E > is a hy-

pertree HD=< T, χ,λ > which satisﬁes the following

conditions:

1. For each edge h ∈ E , there exists p ∈ vertices(T)

such that vars(h) ⊆ χ(p). We say that p covers h.

2. For each variable v ∈ V, the set {p ∈

vertices(T)|v ∈ χ(p)} induces a connected sub-

tree of T.

3. For each vertex p ∈ vertices(T),χ(p) ⊆

var(λ(p)).

The width of Generalized hypertree decomposition

< T,χ, λ > is max

p∈vertices(T)

|λ(p)|. The generalized

hypertree width: Ghw(H ) of hypergraph is the mini-

mum width over all its generalized hypertree decom-

positions

Deﬁnition 6 (Hypertree Decomposition (Gottlob

et al., 2001)). A hypertree decomposition is a gener-

alized hypertree decomposition decomposition which

ICAART 2009 - International Conference on Agents and Artificial Intelligence

additionally satisﬁes the following special condition:

For each vertex p ∈ vertices(T),

var(λ(p))

χ(T

) ⊆ χ(p).

The width of a hypertree decomposition <

T,χ,λ > is max

p∈vertices(T)

|λ(p)|. The hypertree

width: hw(H ) of hypergraph is the minimum width

over all its hypertree decompositions.

Deﬁnition 7 A hyperedge h of a hypergraph H =<

V,E > is strongly covered in HD=< T,χ,λ > if there

exists p ∈ vertices(T) such that the vertices in h are

contained in χ(p) and h ∈ λ(p).

Deﬁnition 8 A hypertree decomposition < T, χ,λ >

of a hypergraph H =< V,E > is a complete hypertree

decomposition if every hyperedge h of H =< V,E >

is strongly covered in HD =< T,χ,λ >.

3 THE HEURISTIC ALEA

To compute hypertreedecomposition, two approaches

are proposed in the literature : the exact meth-

ods (Gottlob et al., 1999b; Harvey and Ghose, 2003)

and the heuristic ones (Korimort, 2003; Musliu and

Schafhauser, 2007; Dermaku et al., 2005). In this sec-

tion, we introduce our new heuristic for computing

hypertree decomposition.

3.1 Theoretical Consideration

First we consider a particular hypertree decomposi-

tion which satisﬁes the following conditions

1. For each edge h ∈ E , there exists p ∈ vertices(T)

such that var(h) ⊆ χ(p). We say that p covers h.

2. For each variable v ∈ V, the set {p ∈

vertices(T)|v ∈ χ(p)} induces a connected sub-

tree of T.

3. For each vertex p ∈

vertices(T), χ(p) = var(λ(p)).

The ﬁrst two conditions are the same as the ﬁrst

two conditions of the hypertree decomposition. The

condition 3 is a particular case of the condition of the

hypertree decomposition which forces the χ labels of

each node p of T to be equal to var(λ(p)). In addi-

tion, this same condition 3 satisﬁes trivially the con-

dition 4 of the hypertree decomposition.

3.2 Description of Alea

BE and DBE are both two extensions of the BE

heuristic for tree decomposition. Unlike BE and DBE

the heuristic Alea computes a hypertree without us-

ing the tree decomposition processing. We look for

only the λ − labels since the χ − labels are obtained

for free (At each node p, the set χ(p) is the union

of vertices of the each hyperedge c , for c ∈ λ(p) .

Our goal is to obtain a hypertree decomposition with

a small width, thus we have to minimize the size of

the λ− labels.

The algorithm 1 Gen-decomposition gives the main

generic procedure for computing a hypertree decom-

position of a given hypergraph H . This algorithm be-

gins by determining the ﬁrst separator or the root of

the hypertree (Line 4 of the algorithm 1) for a given

hypergraph H . Once the ﬁrst separator is deﬁned a

node root for the hypertree is created. In Line 8 of

the algorithm 1, the procedure Decompose is called

to decompose the resulting hypergraph.

Algorithm 1: Gen-decomposition(H ).

1: Input: a given hypergraph H

2: Output: its hypertree H T

3: Begin

4: Find-ﬁrst-separator( H , MinSep);

5: create− node(root,HT);

6: λ(root) = λ(MinSep);

7: χ(root) = vars(λ(MinSep);

8: Decompose(H , MinSep,root );

9: End.

This algorithm is generic because it can use the

same procedure Decompose and different procedures

to ﬁnd the ﬁrst separator. In this paper we consider

the heuristic Alea to determinate the ﬁrst separator.

This heuristic considers as a ﬁrst separator, the ﬁrst

constraint introduced by the user or the ﬁrst constraint

in the ﬁle giving the description of a CSP instance. In

the sequel, we describe the procedure Decompose.

3.2.1 The Procedure Decompose

The procedure Decompose (see algorithm 2) consid-

ers a given hypergraph or a subhypergraph Comp (set

of hyperedges from E), the current separator Sep and

the last node p in the hypertree as parameters. An ap-

propriate separator for a component (sub-hypergraph)

must be a minimal size separator composed of a set of

hyperedges of the given component covering the vari-

ables of intersection between the variables of the last

separator and those of the component. Decompose

uses an auxiliary function and an auxiliary procedure

named respectively separate and newsep.

The function Separate(Comp, sep): Given a com-

ponent Comp and a set of hyperedges sep, separate

removes the hyperedges of sep from the hyperedges

HYPERTREE DECOMPOSITION FOR SOLVING CONSTRAINT SATISFACTION PROBLEMS

Algorithm 2: Decompose (Comp, Sep, p ).

1: Input: Comp a component to be decompose ,

Sep the current separator to use for decomposing

Comp, p the last node in the hypertree;

2: Output: a node p’ son of p;

3: Begin

4: Components := separate(Comp,Sep);

5: for each component C

∈ Components do

6: new-sep(C

, Sep, NewSep);

7: create-node(p’);

8: λ(p

′

) = λ(NewSep);

9: χ(p

′

) = vars(λ(NewSep);

10: connect (p’, p ); // p’ is the son of p in the hy-

pertree;

11: if (isdecomposable) then

12: Decompose(C

, NewSep,p’ );

13: end if

14: end for

15: End.

of comp in order to obtain some connected compo-

nents. For each of these connected components, the

procedureDecomposecalls the procedure new-sep to

determine the new separator to be used again for de-

composing this current component (see algorithm 2,

line 4-7). This new separator is computed by using

the set cover techniques (Computing a minimal set

of hyperedges covering the variables of intersection

between the variables of the last node and the vari-

ables of the component) and by respecting the rules

of hypertree decomposition as well. This procedure

Decompose is recursively called. This process stops

when all the components become not decomposable.

A component is not decomposable if the minimal set

of hyperedges required to cover the intersection of

its variables with the variables of the last separator

is equal to the number of all its hyperedges or if the

size of the component is 1 (the component has only

one hyperedge).

3.2.2 The Procedure New-sep

This procedure ﬁnds a set Y of all hyperedges in the

component Comp that have at least one variable in the

last node Sep, (see algorithm 3) and calls the proce-

dure minrecovering-sep which mainly minimizes the

size of Y.

3.2.3 The Procedure Minrecovering-sep

Given a set of variables X and a set of hyperedges

E, this procedure looks for the minimal set of

hyperedges c (c ∈ E)) , covering a maximum number

Algorithm 3: New-sep (Comp, Sep, Bestsep).

1: Input: Comp a component, Sep is the last sepa-

rator used

2: Output: Bestsep a separator for decomposing

Comp and isdecomposable a boolean;

3: Begin

4: NSep := φ; // Nsep : set of hyperedges that have

at least one variable in Sep

5: for each hyperedge c ∈ hyperedges(Comp) do

6: if (vars(Sep) ∩ vars(c)) 6= φ) then

7: NSep := NSep ∪ c;

8: end if

9: end for

10: End.

11: isdecomposable=true;

12: minrecovering-sep(Comp,vars(NSep),Bestsep,

isdecomposable);

13: End.

of variables in X. Then it removes these covered vari-

ables from the set X, removes the hyperedge c from

E and takes c as a member of the covering set. The

same process is applied for the induced sets X and E

until X becomes empty. The algorithm is deliberately

omitted because it is classical. This algorithm

returns Bestsep and the boolean isdecomposable

witch is true if the component is decomposable.

Notice that our heuristic is different from the one

in (Korimort, 2003) in the sense that we do not

add a special hyperedge to a component to ﬁnd an

appropriate separator for a component. In addition,

our heuristic gives by construction a complete de-

composition in which each hyperedge appears only

once in node p of the resulting hypertree.

The ﬁgure 1(a) shows a given hypergraph and the

ﬁgure 1(b)) gives its hypertree decomposition using

Alea.

f g

p s

c11

c10

b ca

{c7, c9}

{c1}

{c2}

{c3, c4}

{c5} {c10, c11} {c8}

a) A hypergraph H

b) Its hypertree decomposition using Alea

Figure 1: A hypergraph and its hypertree deccomposition.

ICAART 2009 - International Conference on Agents and Artificial Intelligence

Table 1: Grid2d family: comparing exact and heuristic methods.

Instances Size BE DBE Alea dkd

|V| |E| htw w t w t w t w t

grid2d 10 50 50 4 5 0 6 0 6 0 4 0

grid2d 15 113 112 6 8 0 9 0 9 0 6 3

grid2d 20 200 200 7 12 0 11 28 10 2 7 3140

grid2d 25 313 312 9 15 3 15 43 21 0 10 200

grid2d 30 450 450 11 19 7 20 8 14 1,6 13 1566

grid2d 35 613 612 12 23 15 23 73 26 2 15 1905

grid2d 40 800 800 14 26 28 25 31 19 6 17 253

grid2d 45 1013 1012 16 31 51 30 56 30 11 21 26

grid2d 50 1250 1250 17 33 86 32 94 21 39 24 2786

grid2d 60 1800 1800 21 41 204 34 178 33 72 31 3940

3.3 Correctness of Our Heuristic

In this section, we prove that our heuristic computes

a hypertree decomposition.

Proof 1 1. The condition 1 is satisﬁed due to the fact

that in our approch, all hyperedges appear in the

hypertree and since χ(p) = vars(λ(p)) for each

node p in the hypertree.

2. To prove the condition 2, two cases are to con-

sider.

• case1: p

is the parent of p

and p

is the par-

ent of p

. We have to prove that the proposi-

tion x ∈ χ(p

),x ∈ χ(p

) and x /∈ chi(p

) is not

possible. In fact if x ∈ χ(p

), according to the

procedure Decompose either x ∈ χ(p

) and so

mutch the better. Either x /∈ χ(p

) and then ac-

cording to the procedure Decompose x /∈ χ(p

• case2: p

and p

are the sons of p

. We have

to prove that the proposition x ∈ χ(p

) and

x ∈ χ(p

) and x /∈ χ(p

) is not possible. Let

be x ∈ p

then either x ∈ chi(p

) and so mutch

the better. Either x /∈ χ(p

) an then x /∈ χ(p

)

3. The condition 3 is trivially satisﬁed.

4. The condition 4 of the deﬁnition 5 is trivially

satisﬁed because for each vertex p ∈ vertices(T),

var(λ(p))

χ(T

) ⊆ χ(p) = var(λ(p)).

(becauseχ(p) = var(λ(p))).

4 EXPERIMENTS

In this section we report the computational results

obtained with implementation of our heuristic Alea

when we applied it to some different classes of

hypergraphs (Ganzow et al., 2005). We compare our

results with the results obtained from evaluations of

BE and DBE. The tests for BE and DBE have been

done with three variable ordering heuristics MCS

(Maximum Cardinality Search), MIW (Minimum

Induced Width) and MF(Min Fill). But we report

here only the best obtained results in each case.

Nevertheless, MIW is in the general the best variable

ordering. All the experiments were performed on

PENTIUM 4,2.4 GHZ, 512 MO of RAM and running

under OPENSUSE Linux 10.2.

• Dimensional Grids Benchmarks: The ﬁrst class

of benchmarks are hypergraphs extracted from

two dimensional grids. The results obtained are

reported in the table 1, where the ﬁrst column con-

tains the name of the example. The columns 2 and

3 contain respectivelythe number of variables and

the number of hyperedges. The column 4 contains

the hypertree width if this is known. Then there

are two columns for each heuristic (BE, DBE and

Alea). The ﬁrst one contains the minimal width

and the second one gives the runtime in seconds.

For this class of hypergraphs, in (Gottlob and

Samer, 2007) the authors observe that even if

the optimal decompositions can be constructed by

hand, their algorithm det-k-decomp seems suffer

to compute a hypertree decomposition. They ob-

serve also that opt-k-decomp cannot solve any of

them within a timeout of one hour. Notice that

Alea Solves all the problems with a width close

to those obtained with det-k-decomp. In addition,

the CPU time is smaller with Alea. These results

are reported in the table 1. The results for det-k-

decomp and opt-k-decomp are taken from (Gott-

lob and Samer, 2007).

• The Dubois Benchmarks: In the table 2 we re-

port the results obtained with the Dubois bench-

marks which is a particular benchmark of SAT

problems. Alea is equivalent to BE and DBE.

HYPERTREE DECOMPOSITION FOR SOLVING CONSTRAINT SATISFACTION PROBLEMS

Table 2: Dubois family: Comparing Alea, BE and DBE.

Instances |V| |E| BE DBE Alea

Dubois20 60 40 2 2 2

Dubois21 63 42 2 2 2

Dubois22 66 44 2 2 2

Dubois23 69 46 2 2 2

Dubois24 72 192 2 2 2

Dubois25 75 200 2 2 2

Dubois26 78 208 2 2 2

Dubois27 81 216 2 2 2

Dubois28 84 224 2 2 2

Dubois29 87 232 2 2 2

Dubois30 90 240 2 2 2

Dubois50 150 400 2 2 2

Dubois100 300 800 2 2 2

5 SOLVING ACYCLIC CSP

In order to show the practical interest of our approach,

we experiment here the resolution of a given CSP

after its transformation into a hypertree. Given a

CSP and its structutal representation into an hyper-

graph G =< V,E > we ﬁrst compute its hypertree

< T, χ,λ > using Alea. Then, the hypertree is trans-

formed into an equivalent acyclic CSP . Finally, the

resulting CSP is solved using the Acyclic Solving al-

gorithm (Dechter et al., 1988)

5.1 Building and Equivalent Acyclic

CSP

Given a CSP P = (X,D,C,R) and its hypertree de-

composition < T,χ,λ >, the equivalent acyclic CSP

P = (X

′

D,C

′

) is obtained as follows: X

′

= X ,

′

= D , C

′

= {χ(p)|p ∈ vertices(T)} and R

′

= {⋊⋉

(λ(p)[χ(p)]|p ∈ vertices(T)} where ⋊⋉ represents the

join operation and [X] represents the projection oper-

ator on the set X of variables.

5.2 Acycling Solving Algorithm

The ﬁrst algorithm proposed is the tree solv-

ing(Dechter et al., 1988) which solves acyclic binary

CSPs in polynomial time The generalization of this

algorithm for solving non binary CSP gave rise to the

Acyclic Solving algorithm (Dechter, 2003) . This al-

gorithm begins by ordering the nodes of the hypertree

such that every node parent precedes its sons( Ligne 4

of 4). Then, it treats successively all the nodes com-

ing from the leaves to the root. For each node p of

T, it realizes a semi jointure on itself and its parent

in order to ﬁlter the parent relation ( Lignes 6-12 4)

. When the root is treated without making empty any

relation, the solution of the CSP is determinated in a

backtrack free manner (Lignes 14-16 of 4).

Algorithm 4: Acyclic Solving (Comp, Last-sep, Sep,

p ).

1: Entres: P = (X,D,C,R), R = {R

,... ,R

an hypertree < T,χ, λ >

2: Sortie : Veriﬁes the consistency and looks for in

the positive case.

3: Dbut

4: d = (R

,... ,R

) an ordering where each rela-

tion preceeds its sons in the rooted hypertree T.

5: for i= t to 2 do

6: for each edge (j, k) , k ¡ j, in the hypertree do

7: R

= (R

∝ R

)[C

]

8: if R

0 then

9: Exit; The problem is inconsistent.

10: end if

11: end for

12: end for

13: /* The CSP is consistent, look for its solution

14: Select a tuple from R1.

15: for i = 2 t do

16: Select a tuple from Ri consistent withe previ-

ous choosen tuples.

17: end for

Table 3: Dubois family: Comparing Acyclic Solving and

nFC2-MRV.

Instances |V| |E| Alea BE nFC2-MRV

Dubois20 60 40 0,23 0,31 419

Dubois21 63 42 0,24 0,34 896

Dubois22 66 44 0,24 0,34 1916

Dubois23 69 46 0,29 0,33 15026

Dubois24 72 48 0,29 0,37 -

Dubois25 75 50 0,27 0,34 -

Dubois26 78 52 0,29 0,36 -

Dubois27 81 54 0,31 0,39 -

Dubois28 84 56 0,36 0,41 -

Dubois29 87 58 0,32 0,42 -

Dubois30 90 60 0,35 0,43 -

5.3 Experimental Evaluation

Our approach has been tested on 11 SAT prob-

lems from the Dubois familly. For each prob-

lem, we compare the enumerative algorithm nFC2-

mrv (C.Bessi`ere and Larossa, 2002) with the Acyclic

Solving algorithm 4 applied to the hypertree decom-

positions derived from BE and Alea heuristics. Our

experiments are reported in the table 3. We clearly ob-

serve that Acyclic Solving algorithm outperforms the

ICAART 2009 - International Conference on Agents and Artificial Intelligence

CPU time. The symbol − signiﬁes that nFC2-MRV

has been interruped after 5 hours of execution time.

We also observe that the results obtained by Alea are

slightly better that those obtained by BE. This is due

to the fact that no jointure operations has been done

by using Alea.

Finally the ﬁgures ﬁgures 2 and 3 show the prac-

tical interest of a polynomial algorithm comparing to

an exponential one.

Figure 2: Solving with Acyclic Solving, BE and Alea.

Figure 3: Solving with nFC2-MRV.

6 CONCLUSIONS

We have studied and tested an approach to solve CSP

by decomposition.

First, we proposed the heuristic Alea to compute

the hypertree decomposition. Some experiments have

shown that Alea is comparable and sometimes bet-

ter on certain problems than BE (the most successful

heuristic and DBE). Then we have studied and imple-

mented an acyclic solving (Dechter, 2003) algorithm

to solve a given CSP after it was transformed into a

hypertree decomposition. Our heuristic ”Alea” is par-

ticular as it does not require the projection operations.

As a consequence, execution time to solve a CSP us-

ing Alea is slightly lower than with BE.

We have shown the interest of this approach on

a particular family of problems. A work currently

in progress consists in the development of a parallel

acyclic solving algorithm with a judicious strategy for

the saving of data. Indeed besides the CPU time, one

of the major disadvantages of the Acyclic Solving is

the use of a large space memory.

REFERENCES

C.Bessi`ere, P. Meseguer, C. F. and Larossa, J. (2002). On

forward checking for non binary constraint satisfac-

tion. Artiﬁcial Intelligence, 141:205–224.

Cohen, D., Jeavons, P., and Gyssens, M. (2005). A uni-

ﬁed theory of structural tractability for constraint sat-

isfaction problems and spread cut decomposition. In

Proceedings of IJCAI 2005, pages 72–77.

Dechter, R. (2003). Constraint Processing. Morgan Kauf-

mann.

Dechter, R., Dechter, A., and Pearl, J. (1988). Optimization

in constraint-networks. In Proceedings of the Con-

ference on Inﬂuence Diagrams for Decision Analysis:

Inference and prediction, Berkeley.

Dechter, R. and Pearl, J. (1989). Tree clustering for con-

straint networks. Artiﬁcial Intelligence, 38:353–366.

Dermaku, A., Ganzow, T., Gottlob, G., McMahan, B., Mus-

liu, N., and Samer, M. (2005). Heuristic methods for

hypertree decompositions. Technical report, DBAI-R.

Freuder, E. C. (1982). A sufﬁcient condition for backtrack-

free search. Journal of the Association for Computing

Machinery, 29:24–32.

Ganzow, T., Gottlob, G .and Musliu, N., and Samer, M.

(2005). A csp hypergraph library. Technical report,

DBAI-TR-2005-50, Technische Universitt Wien.

Gottlob, G., Grohe, M., and Musliu, N. (2005). Hyper-

tree decomposition: structure, algorithms and applica-

tions. In Proceeding of 31 st International workshop

WG, Metz.

Gottlob, G., Leone, N., and Scarcello, F. (1999a). A com-

parison of structural csp decomposition methods. In

Proceedings of IJCAI’99, pages 394–399.

Gottlob, G., Leone, N., and Scarcello, F. (1999b). On

tractable queries and constraints. In Proceedings of

DEXA’99.

Gottlob, G., Leone, N., and Scarcello, F. (2000). A compar-

ison of structural csp decomposition methods. Artiﬁ-

cial Intelligence, 124:243–282.

Gottlob, G., Leone, N., and Scarcello, F. (2001). Hypertree

decompositions: A survey. In Proceedings of MFCS

’01, pages 37–57.

Gottlob, G. and Samer, M. (2007). A backtraching based

algorithm for computing hypertree decompositions.

arXivics.DS 0701083vl 14 Jan 2007.

Gyssens, M., Jeavons, P. G., and Cohen, D. A. (1994).

Decomposing constraint satisfaction problems using

database techniques. Artiﬁcial Intelligence, 66:57–89.

Harvey, P. and Ghose, A. (2003). Reducing redundancy in

the hypertree decomposition scheme. In Proceding of

ICTAI’03, pages 548–555, Montreal.

HYPERTREE DECOMPOSITION FOR SOLVING CONSTRAINT SATISFACTION PROBLEMS

Jeavons, P. G., A, C. D., and Gyssens, M. (1994). A struc-

tural decomposition for hypergraphs. Contemporary

Mathematics, 178:161–177.

Korimort, T. (April 2003). Heuristic decomposition Hyper-

tree Decomposition. PhD thesis, Vienna University of

Technology.

McMahan, B. (2003). Bucket elimination and hypertree de-

compositions. Technical report, Implementation re-

port, institute of information systems (DBAI), TU, Vi-

enna.

Montanari, U. (1974). Networks of constraints: Fundamen-

tal properties and applications to pictures processing.

Information Sciences, 7:95–132.

Musliu, N. and Schafhauser, W. (2007). Genetic algorithms

for generalized hypertree decompositions. European

Journal of Industrial Engineering, 1(3):317–340.

ICAART 2009 - International Conference on Agents and Artificial Intelligence