Arc Consistency Projection: A New
Generalization Relation for Graphs
Michel Liquiere
LIRMM,
161 Rue ada,
34392 Montpellier cedex 5,
France
Liquiere@lirmm.fr
Abstract. The projection problem (conceptual graph projection, homo-
morphism, injective morphism, θ-subsumption, OI-subsumption) is cru-
cial to the efficiency of relational learning systems. How to manage this
complexity has motivated numerous studies on learning biases, restrict-
ing the size and/or the number of hypotheses explored. The approach
suggested in this paper advocates a projection operator based on the
classical arc consistency algorithm used in constraint satisfaction prob-
lems. This projection method has the required properties : polynomiality,
local validation, parallelization, structural interpretation. Using the arc
consistency projection, we found a generalization operator between la-
beled graphs. Such an operator gives the structure of the classification
space which is a concept lattice.
1 Introduction
The complexity of the computation of the generality relation between two rela-
tional descriptions, is a crucial problem. For conceptual graphs [1] this operation
is named projection. Such an operation is linked to a classical problem in the
graph community: the search for an homomorphism between two graphs. As
stated in [2], “the elementary reasoning operation, projection is a kind of graph
homomorphism that preserves the partial order defined on labels”. The search
for an homomorphism between a tree and a graph is polynomial but between
general graphs, the problem is NP complete [3]. In conceptual graph community,
different algorithms are proposed for the projection problem [4,5,6].
From another point of view, Inductive Logic Programming systems (ILP)
commonly used a generality relation, a decidable restriction of logical implica-
tion named θ-subsumption. The homomorphism is also directly linked to the
θ-subsumption operation [7]. In machine learning, the complexity of this opera-
tion has motivated the use of learning biases: syntactic biases (trees [8], specific
graph [9,10]), efficient implementation [7] and approximation of θ-subsumption
[11].
Finally, the homomorphism is also linked to the classical constraint program-
ming resolution (CSP) [12]. This final link gives an interesting algorithmic point
U. Priss, S. Polovina, and R. Hill (Eds.): ICCS 2007, LNAI 4604, pp. 333–346, 2007.
c
Springer-Verlag Berlin Heidelberg 2007
334 M. Liquiere
of view since CSP community has many results which improve the resolution
algorithm.
In this paper, we propose to use a part of these three domains for a classical
unsupervised machine learning problem. We represent each example by a labeled
graph. We use a new generality relation named AC-projection based on the arc
consistency algorithm [12] and we prove that the search space, for a relational
machine learning classification problem, is a concept lattice [13].
2 A New Projection: AC-Projection
Any constraint satisfaction problem can be viewed as a “network” of variables
and constraints. In this network each variable is connected to the constraints that
involve it and each constraint is connected to the variables it involves. Among
backtracking based algorithms for constraint satisfaction problems, algorithm
employing constraint propagation, like forward checking and arc consistency [12],
have had the most practical impact. These algorithms use constraint propagation
(arc consistency) during a search to prune inconsistent values from the domains
of the uninstantiated variables.
2.1 AC-Projection and Arc Consistency
In this paragraph we present the arc consistency using a graph notation. For
other presentations see the books [3,12].
Notation. For a labelled directed graph, named digraph in this paper, G, we
note V (G) the set of vertices of G, A(G)thesetofarcsofG,L(G)thesetof
labels of G. For a vertex x ∈ V (G)wenotel(x ) ∈ L(G)thelabelofx,N(x)
the set of all the neighbors of x, P (x) ⊆ N(x) the predecessors of x and S(x) ⊆
N(x) the successors of x.
For a finite set S we note 2
S
the set of all subsets of S (power set). In this
paragraph we study some important properties of the arc consistency.
Definition 1 (labeling). Let G
1
and G
2
be two digraphs. We named labeling
from G
1
into G
2
a mapping I:V(G
1
) → 2
V (G
2
)
|∀x ∈ V(G
1
), ∀ y ∈I(x),
l(x)=l(y).
Thus for a vertex x ∈ V(G
1
), I(x) is a set of vertices of G
2
with the same label
l(x). We can think of I(x) as the set of “possible images” of the vertex x in G
2
.
This first labeling is trivial but can be refined using the neighborhood relations
between vertices.
Definition 2 (∼). Let G be a digraph, V
1
⊆ V(G), V
2
⊆ V(G).
We note V
1
∼ V
2
iff
1) ∀x
k
∈ V
1
∃y
p
∈ V
2
| (x
k
,y
p
) ∈ A(G)
2) ∀y
q
∈ V
2
∃x
m
∈ V
1
| (x
m
,y
q
) ∈ A(G).
In this definition we give a direct relation between two sets of vertices V
1
and
V
2
. So for each vertex x
k
of V
1
there is at least one vertex y
p
of V
2
which is a
Arc Consistency Projection: A New Generalization Relation for Graphs 335
neighbor of x
k
:((x
k
,y
p
) ∈A(G)) and all vertices of V
2
are a neighbor of, at least,
one vertex of V
1
(oriented condition). This is not a one to one relation like the
subgraph isomorphism.
Definition 3 (Consistency for one arc). Let G
1
and G
2
be two digraphs. We
say that a labeling I:V(G
1
) → 2
V (G
2
)
is consistent with an arc (x, y) ∈A(G
1
),
iff I(x) ∼ I(y).
In the example of Figure 1, a vertex is designated by a letter and a number: the
letter is the label of the vertex and the number is only an identification number.
In this example the labeling I:I(a
0
)={a
4
,a
10
} and I(b
1
)={b
5
,b
9
}, is consistent
with the edge (a
0
,b
1
)sinceI(a
0
)∼I(b
1
).
Definition 4 (AC-projection ). Let G
1
and G
2
be two digraphs. A labeling
I from G
1
into G
2
is an AC-projection iff I is consistent with all the arcs e ∈
A(G
1
). We note it G
1
G
2
The name “AC-projection” comes from the classical AC (arc consistency) used
in [12].
a b c
c b a
c
a b
c
c
45 7
012
3
89 10
6
G
1
G
2
Fig. 1. AC-Projection: Example
Consider the labelling I(a
0
):{a
4
,a
10
}, I(b
1
):{b
5
,b
9
}, I(c
2
):{c
6
,c
7
,c
8
},I(c
3
):
{c
7
,c
8
}.WeverifyI(a
0
)∼ I(b
1
), I(b
1
)∼I(c
2
), I(b
1
)∼I(c
3
), I(c
3
)∼I(a
0
).
Then I is an AC-projection from G
1
into G
2
since I is a labelling consistent
with all arcs of G
1
.
2.2 AC-Projection Properties
We have defined a new mapping relation between graphs. In this paragraph we
study the properties of this relation (complexity, interpretation).
We recall the classical homomorphism definition for digraphs.
336 M. Liquiere
Definition 5 (homomorphi sm → ). A homomorphism of a digraph G
1
to a
digraph G
2
is a mapping of the vertex sets f:V(G
1
) → V(G
2
) which preserves
the arcs and labels, i.e such that (x,y) ∈ A(G
1
) ⇒ (f(x),f(y)) ∈ A(G
2
)and∀x,
l(x)=l(f(x)).
Notation: G
1
→ G
2
Note that for digraph (f(x),f(y))∈A(G
2
)impliesthatf(x)= f(y), since each edge
of A(G
2
) consists of two distinct elements.
We have the following proposition which links the AC-projection to the Ho-
momorphism.
Proposition 1. For two digraphs G
1
and G
2
,ifG
1
→ G
2
then G
1
G
2
.
Proof. See [3]
This proposition is the foundation of many CSP resolution methods. These meth-
ods are based on the classical arc consistency algorithm AC1 used in CSP,
which has been improved (AC2 ... AC5), the actual minimal complexity is:
O(ed
2
)wheree is the number of arcs and d thesizeofthelargestdomain
[12].
In our case, the size of the largest domain is the size of the largest subset of
nodes with the same label. So an AC-projection between two digraphs can be
computed in polynomial time.
2.3 AC-Projection Algorithm
We give a simple AC-projection algorithm for digraphs (based on AC1 algorithm
[12]).
This algorithm Arc-Consistency takes two digraphs G
1
, G
2
and tests if there
is an AC-projection from G
1
into G
2
. It begins by the creation of a first rough
labeling I and reduces, for each vertex x, the given lists I(x) to consistent lists
using the procedure ReviseArc.
The consistency check fails if some I(x) becomes empty; otherwise the con-
sistency check succeeds and the algorithm gives the labeling I whichisanAC-
projection G
1
G
2
.
Procedure: ReviseArc
Data:Anarc(x,y)∈ V(G
1
)
Data: A labeling I from G
1
into G
2
Data: A digraph G
2
Result: A new labeling I
from G
1
into G
2
I
:= I ;
I
(x):= I(x) - {x’ ∈ V(G
2
) | ∃ y’ ∈I(y) with (x’,y’) ∈ A(G
2
)};
I
(y):=I(y) - {y’ ∈ V(G
2
) | ∃ x’ ∈I(x) with (x’,y’) ∈ A(G
2
)};
return I
Arc Consistency Projection: A New Generalization Relation for Graphs 337
Procedure: Arc-Consistency
Data: Two digraphs G
1
and G
2
Result: An AC-projection I from G
1
into G
2
if there is one else an empty
set ∅
// Initialisation
for x ∈ V(G
1
) do
I(x)={y ∈ V (G
2
) | l(x)=l(y))} ;
end
S := A(G
1
);
while S = ∅ do
Choose an arc (x,y) from S; // In general the first element of S
I
:=ReviseArc((x,y),I,G
2
);
//If for one vertex x ∈ V(G
1
)wehaveI
(x)= ∅ then there is no arc
consistency
if (I
(x)=∅) or (I
(y)=∅) then
return ∅;
end
// I
is consistent now with the arc (x, y); but it can be non-consistent
with some other previously tested arcs so we have to verify and change
(if necessary), the consistency of all these arcs.
if I(x) = I
(x) then
S := S
{(x
,y
) ∈ V (G
1
) | x
= x or y
= x};
end
if I(y) = I
(y) then
S := S
{(x
,y
) ∈ V (G
1
) | x
= y or y
= y};
end
Remove (x,y) from S;
I:=I
;
end
return I;
The Arc-Consistency algorithm has a polynomial time complexity [3,12] and
gives, if there is one, an AC-projection I from G
1
into G
2
verifying: for all
AC-projection I
from G
1
into G
2
,wehave∀ x ∈ V(G
1
), I
(x) ⊆I(x) [3].
3 AC-Projection and Machine Learning
In[10],wehavestudiedtheconstructionof a concept lattice, where the extension
part is a subset of the set of example but where the intension part is described
by a digraph. In the context of machine learning, the automatic bottom up
construction of such a hierarchy can be viewed as an unsupervised conceptual
classification method. In this paper the generalization partial order was based
on homomorphism relation between digraph. To deal with the homomorphism
complexity, we proposed a class of digraph with a polynomial homomorphism
operation. This limit the generality of the description language.