From: FLAIRS-01 Proceedings. Copyright © 2001, AAAI (www.aaai.org). All rights reserved.
Binary Representations
Wanlin
Pang
Institute for Information Technology
National Research Council of Canada
Ottawa, Ontario, Canada K1A0R6
Email: ’ Wanlin.Pang~nrc.ca
Abstract
It is well knownthat a non-binaryconstraint satisfaction problem(CSP) can be transformed into an equivalent binary CSPusing the dual graph transformation.
It is also knownthat the dual graph transformation is
impractical in some cases where the CSPhas a large
numberof constraints and/or the constraints have a
larger numberof satisfying tuples. However,little work
has been done on improving transformation methods.
In this paper, we introduce a constraint representative
graph called w-graph and present a new transformation methods based on the w-graph. Weshow that
the w-graphbased transformation is a generalization
of the dual graph tra~asformationand it overcomescertain weaknesses of the dual graph transformation in
manycases.
Introduction
A non-binary constraint satisfaction problem (CSP) can
either be solved directly or transformed into a binary
one and then solved by using binary CSP techniques.
There have been direct solution techniques developed
for solving non-binary CSPs effectively
(Mackworth
1977; Van Hentenryck 1989; Pang & Goodwin 1996;
1997). A potential advantage of transformation approach is that binary CSP has been extensively studied which has generated great deal of knowledge about
the theory and practice of solving it. Recently, Bacchus and van Beek (Bacchus & van Beek 1998) reported
their studies on the transformation from non-binary to
binary CSPs and showed that for the CSPs with some
particular properties, that is, the numberof constraints
is low relative to the number of variables and the constraints are restrictive,
even the standard dual graph
based transformation method could be more efficient
than solving a non-binary CSP directly by a search
method such as adapted forward checking. Unfortunately, little work has been done on transformation itself. The question of whether there exist other transformation approaches or how to improve existing transformation remains unexplored. This paper addresses this
problem.
Copyright(~) 2001, AmericanAssociation for Artificial Intelligence (www.aaai.org).All rights reserved.
for General CSPs
Scott
D. Goodwin
Department of ComputerScience
University of Regina
Regina, Saskatchewan, Canada $4S 0A2
Emaih goodwin~cs.uregina.ca
We present a new transformation
method based on
a constraint representative graph called w-graph. An
w-graph is a generalization of dual graph in that a dual
graph is an w-graph whereas the converse does not necessarily hold. Given a CSP (binary or non-binary),
can construct an w-graph that is simpler than the dual
graph in terms of cyclicity, width, and numberof nodes.
Accordingly, a non-binary CSP can be transformed to
an equivalent binary one based on the w-graph, and the
binary CSPobtained from the w-graph based transformation is usually simpler than the one obtained from
the dual graph based transformation in terms of the
number of variables and/or the size of domains. In
particular, the number of variables in the binary CSP
obtained from the w-graph transformation does not exceed the numberof variables in the original CSP. For the
class of CSPs where the number of constraints are small
and the constraints are tight, according to (Bacchus
& van Beek 1998), the dual graph transformation performs well and thus the w-graph transformation could
be better. For CSPs with large number of constraints
and large numberof satisfying tuples in the constraints,
the w-graph provides us with a tool to exploit the constraint structure and to transform a non-binary CSPto
a binary one based on its structure.
Preliminaries
Wefirst define constraint satisfaction problems (CSPs)
and then briefly review related work on binary representations for general CSPs.
Constraint Satisfaction
Problems
A constraint satisfaction problem (CSP) is a structure
(X,D, V,S). Here, X = {X1, X~., ..., Xn} is a set of
variables that may take on values from a set of domains
D = {01, D2, ...,
On}, 1and Y = {V1,V2,...,Vm}
is a family of ordered subsets of X called constraint
schemes. Each Vi = {Xil,Xi2,...,Xir,}
is associated
with a set of tuples Si C_ Di~ × Di~ x ... x Di,, called
constraint instance, and S ={ $1, $2, . . . , Srn } is a family of such constraint instances. Together, an ordered
1Throughoutthis paper, we assume that Vi, j(V~ 6 V
Vj E V A i ~ j =~Vi ~ V) A Vj g Vi ).
KNOWLEDGE
REPRESENTATION355
pair (~, Si) is constraint orrel ation which per mits
the variables in r~ to take only value combinations in
Si. A solution is an n-tuple from D1 × D~ × ... × Dn
such that all the constraints are satisfied. The task of
solving a CSPis to find one or all solutions.
A binary GSP is a CSP with unary and binary constraints only, that is, every constraint schemecontains
at most two variables. A CSPwith constraints not limited to unary and binary is referred to as a general CSP.
Example 1 Consider a satisfiability
problem, (X1
X2)A(-~X~
v-~X2)A(X~VX4VXr)A(-~X~
VX4V’~Xr)A
(-~X~V-~X4VXT)A(~XlV-~X4V-~XT)
A(-~X2
V
^ix2 v x4 v x~) ^ (x2 v x4 v-xT)A(X2V-X4
v XT)
(~X~v ~X~v X,) ^ (~X2v ~X4v ~X,) A(X3V
XT)A(XaV-~X5
VXT)A(XaV-~X.
V-~X~)
A(-~XaV
~xT)^ (~x3v ~x5v ~x~)A(Xav ~Xs)A (~X3V
This problem can be formalized as a 3-ary CSP with
7 variables X = {X1, X2, Xa, X4, Xh, Xs, XT} and every variable has the same domain {0, 1}. The problem
has 6 constraints on constraint schemes V1 = {X1, X2},
~ = {xl,
x4,xT},
va= {v2,
v3},
v4= {x2,
x4,XT},
V5 = {X3, 25, X7}, V6 = {23, X6}, and constraint
instances
are thosesetsof tuplesas shownin Figure I. For example,
thereis a constraint
on XI and
X2. The permissible value pairs (01) and (10) represent (X1 V X2) A (-~X1 V -~X2).
Dual Graph Representation
A binary CSPis associated with a simple graph in which
nodes represent variables and edges represent binary
constraints.
A general CSP is associated with a hypergraph in which nodes represent variables and hyperedges represent constraints (see Figure 1). A hypergraph has a dual graph s whose nodes are hyperedges
of the hypergraph and with two nodes joined with an
edge if these two nodes share commonvariables (see
Figure 2).
The dual graph representation transforms a general
CSP to a binary one in which the variables are the
original constraint schemes represented by the nodes of
the dual graph (Dechter and Pearl (Dechter & Pearl
1989)). The domains of the new variables are original
constraint instances. There is a compatible constraint
between two variables if there is an edge between two
nodes in the dual graph.
Example 2 The CSP in Example 1 is associated with
a hypergraph, as in Figure 1, which has a dual graph
as shown in Figure 2. In the binary CSPobtained from
dual graph transformation, the variable set is original
constraint scheme set V = {V~, ~, Va, V4, V~, Vs}. The
domains of the new variables and 10 binary constraints
are shown in Figure 2. For example, the domain of I/1
is {1, 2} representing tuples (01) and (10)in the original constraints on V~ = {X1, X2}. The domain of 1/4 is
{3,4,5} representing tuples (011), (100), and (101)in
the original constraints on V4 = {X2, )(4, Xz}. There
a constraint on the new variables V1 and V4 because
they share a commonvariable X2. The permissible
value pair (14), for example, represents that the tuple
(01) for {X~, X2}and the tuple (100) for {X2, )(4,
are compatible.
Figure 1: A general CSPand its hypergraph
Since constraints are defined as relations, we use some
relational operators, specifically, join and projection.
Let Ci = (~, Si) and Cj = (~, Sj) be two constraints,
ti E Si and tj E Sj two tuples, and Vh a subset of
~. The join of Ci and Cj is a constraint denoted by
Ci t~ Q. The projection of Ci = (V~., Si) on V~ C_ Vi is
a constraint denoted by IIv~(Ci). The projection of ti
on Vh, denoted by ti[Vh], is a tuple consisting of only
the componentsof ti that correspond to variables in Vh.
ti and tj are compatible if ti[V/N V~] -- t~[~ fq V~]. If
tl and tl are compatible, the join ofti and t~, denoted
by ti t~ tj, is a tuple such that (ti t~ tj)[V/] = ti and
(t~ ~ t~)[v~]= t~.
A constraint (Vh, Sh) in a CSP(X, D, V, S) is redundant if its removal does not change the relation represented by the CSP; that is, there is a set of constraints
{(V~,, S~,),...,
(Vp,,S~)} such that Vh U~=~V~,and
Uv~(~L1s~,) c_ s~.
356
FLAIRS-2001
Figure 2: The dual graph and the binary CSP
Join Graph Representation
Several researchers (Gyssens 1986; Dechter & Pearl
1989; Jegou 1993b) have noticed that the dual constraint graph obtained from a general CSPcan be simplified by removing rednndant edges resulting in a join
graph. An edge in a dual graph is redundant if the
variables shared by its two end nodes are also shared
by every nodes along an alternative path between the
2Thedual graph is also called inter graph (Jegou 1993a)
and line graph (Maier 1983).
two end nodes. Since all binary constraints in the new
binary CSPare compatible constraints (i.e., the two tupies assigned to the two constrained new variables agree
on values for all shared original variables), the constraints associated with the redundant edges are also
redundant, and therefore can be removed without affecting solutions to the original CSP. In other words,
the join graph representation transforms a general CSP
to a binary one in which there is no redundant constraints that are in the binary CSPobtained from dual
graph transformation.
Example 3 The CSP in Example 1 is transformed to
a binary one based on a join graph as shown in Figure 3. Notice that 3 redundant constraints on {V3, V4},
{V4, Vh}, and {Vh, Vs} are removed.
11 2} (t2)
{0,],2}
c2o, (V~
(1,,
/
(12)
I I(1,4,~
(II)
I .... k (oo)
I:.’;’;~\02)
(3 1)
11,2.3.
,4.6|
{ .4,
(0. 1}
:7 -
Figure 3: A join graph and the binary CSP
A join graph is minimal if it does not contain redundant edges. A minimal join graph can be obtained
from the dual graph by removing all redundant edges.
Given a hypergraph H = (X, V) and its dual graph
I(H) = (V, L), The following algorithm constructs a
minimal join graph j(H) = (V, J). The time complexity of the algorithm is O(IYp).
min-join-graph(( V, L )
1. begin
2. for each i from m to 2 do
for each j from i - 1 to 1 do
3.
4.
if P)~ = ~ NP]" ~
if3k(k < j, Vh C ~nVk)
5.
6.
L ~ L - {(P~, ~)};
7.
else if 3k(j < k < i, Vh C ~ n Vk
8.
L ~- L - {(V/, 1~)};
9. return (V, L);
10. end
w-Graph
Representation
Given a general CSP (X, D, V,S) and its associated
constraint hypergraph H = (X, V), we can construct
dual graph l(H) = (V, whose nod es are hyperedges
of H and with two nodes 1~ and Vj joined with an edge
(V/,P)) E L if~ NVj ~ 0. jo in gr aph of H i s a
partial dual graph, j(H) = (V, g), where J C L and for
every pair ~, Vj in V, if I~ N Vj¢ $, then there exists
in j(H) a chain ~ -- Vp~, Vp~,..., Vpq = Vj such that
V/n ~ C Vp~n Vp~+~for all 1 _< k < q. As a graph representation of CSPs, we observed that a join graph can
be further simplified by deleting some nodes resulting
in a new representative graph which is called w-graph.
In this section, we define the w-graph and show how to
transform a general CSP to a binary one based on an
w-graph.
w-Graph
The node set of an w-graph is a special set of nodes
of the dual graph called w-cover. An w-cover is a subset of constraint schemes with the property that any
constraint scheme not in the w-cover is covered by two
constraint schemes in the w-cover. More formally,
Definition 1 Given a hypergraph H = (X, V). A subset W C_ V is called an w-cover of X if toW = X and if
for any V~ E V- W, there exist Vi, Vj E W such that
Vk C_ ~ tO Vj. An w-cover W is minimal/f there is no
1~, Vj, V~ E Wsuch that V~ C_ ~ U ~.
For example, given a hypergraph H -- (X, V)
in Figure 1, where X = {X1,X2,Xa,X4,Xh,X6,XT)
and V - {V1,V2,Va,V4,Vh,Vs}.
A subset of V,
W1 = {V2,V3,V4,Vh,Vs} is an w-cover, since for V1 {X1,X2} e V- W, there are V2 = {XI,X4, X~},
V4 = {X2, X4, XT} E W1such that V1 C V~ to V4. The
subset W2= {V~.,V4,Vh,Vs} is a minimal w-cover. Notice that V itself is a w-coverbut not necessarily a minimal w-cover.
For findin~n w-cover we can split__V into two subsets, Wand W, where for every Vk in Wthere is a pair
V~ and V~ in Wsuch that V~ C_ ~ to V~, and there are
no P~, P~, and V~in Wsuch that V~ C V/to V~. Initially,
we move all those V~ from V to W, where there are no
V/,~ in V such that V~ C V/to~. Then we repeat
the following process until V is empty: first we move
all these V~ from V to W, where there are V/,V~ E W
such that V~ C ~ U ~; if V is still not empty, then
we select V/ and Vj in W U V and V~ from V such that
V~ C t~ tO Vj, move~, P~ to Wif they are not in Wyet,
and moveV~ to W. This idea is formalized as an algorithm which is given below. Notice that w-cover(V, ~)
has a parameter V~ which is a node subset we wish to
retain in the w-cover.
~-cowr(V,V’)
1. begin
2. w ~ v’ u { v~l ~V~,l~ e Vs.t. V~_Cpiuv~};
3.
4.
5.
6.
7.
8.
9.
V~V-W;
re_peat
w~-{v~lv~ev,3v,,~ws.t,
v~ c ~ uv~};
V~V-W;
ifV ¢~
find V~, l/) e W to V,V~ E V s.t.
__w~___w
u {t~, ~};
V~_C_¼uV~.;
10.
W ~-- WU {Vk};
11.
V +- V - {1~, V), V}};
12. untilV = ~;
13. return W;
14. end
From an w-cover and a hypergraph, we can construct
an w-graph.
KNOWLEDGE
REPRESENTATION357
Definition 2 Let H = (X, V) be a hypergraph and
an w-cover. An w-linegraph of H = (X, V) is a graph
wl(H) = (W, E), where the node set W is an w-cover
and there is an edge joining two nodes Vi and Vj if
either ~ N Vj ~ 0 or there is Vk E V - W such that
vkc v~urn.
Given a w-linegraph wl(H) = (W, E). For each arc
(~, Vj) ¯ E, we define the are-covered-set of V~. and
~ withrespectto Was vksa(~,V~)= {V~lVk
¯V
W, Vk C ~ U ~}. We define the path-covered-set
as
vksp(~, I~ ) = { VklVk ¯ vksa(Vp, Vq)} where (Vp, Vq)
any edge on any path connecting V/ and Vj.
Definition 3 Let wdH) = (W,E) be an w-linegraph
of a hypergraph H = (X, V). An w-graph of H is a
partial w-linegraph w(H) = (W, F), where F C E
that for any (V/,~) ¯ E, (i). ifVifqr~ # 0, then
there exists in w(H) a chain I,~ = Wl, W~.... , Wq =
such that V~ N V~. C Wk for all 1 < k < q; and (ii).
if vks~(~,v~)# ~, thenvkso(~,~) C-vksp(V,,
A minimal w-graph is an w-graph win(H) = (W, F,~)
where there is no F’ C F,n such that (W, F’) is aa
graph.
For example, given a hypergraph H = ( X, V) as in
Figure 1 and an w-cover W= {V2, V4, V5, Vs}, we can
have an w-linegraph and an w-graph as shown in Figure 4 (A) and (B). Notice that an w-linegraph is
w-graph but not necessarily a minimal w-graph.
(A) Anm-lincgraph
(S) An~-graphandthe binaryCSP
Figure 4: w-graphs and the binary CSP
There are different ways to construct an w-graph for a
hypergraph. For example, we can first find an w-cover,
then construct a sub join graph induced by the w-cover,
and finally construct an w-graph by adding necessary
edges representing those constraints not in the w-cover.
This is the idea of the algorithm w-graph as shown in
the following. Other w-graph algorithms can be found
in (Pang 1998).
w-graph(
( Y,L),
1. begin
2. (W, Jw) ~ subgraph of (V, L) induced by
3. (W, F) ,--- min-join-graph((W, Jw));
4. for each Vk ¯ V - W do
5.
if/~(V~, I~) ¯ F s.t. Vk C V/U
find~,~s.t.VkCViUt~;F~--FU{(Vi,Vj)};
6.
7. return (W,F);
8. end
3S8
FLAIRS-2001
Since V is an w-cover, a join graph is also an w-graph
but not necessarily a minimal w-graph. The w-graph
has somenice properties in terms of degree of cyclicity
and width that are crucial for characterizing the time
complexity of solving a CSP, interested readers are referred to (Pang 1998). For binary representation,
present a proposition here without proof.
Proposition 1 Given a hypergraph H = (X, V), we
can construct an w-graph w(H) = (W,F), such
IWI<IXl, in polynomial time.
w-Graph Transformation
A general CSP is associated with a hypergraph which
has an w-graph (including join graph). From a CSP
(x, o, v, s) andanw-graph
w(n)= (w, F),
form a binary CSP (W, ~°, F , R ), w here e ach ~ ¯
is considered a singleton variable that takes on values
from a set of domains Sw = {3i1~ ¯ W} C S. F is
a set of binary constraint schemes and R is a set of
constraint instances. Each edge Fij = {~, ~ } in F is
associated with a subset R0 = (IIv,(Sij) = IIv~(Sij))
in R where S0 C_ Si t~ Sj such that IIvh (SO) C_ Sh for
all vh¯ vks~(V~
u v~).
In other words, the binary constraint posed on V/and
requests that the instantiation of variables in r~ and
has to satisfy the compatible constraint as well as
the original constraints in arc-covered-set vksa(Vi, Vj).
Weprove that such a binary CSP is equivalent to the
original general CSP; that is, they represent the same
relation on X.
Proposition 2 If (W, w, F , R ) i s a bi nary CSP convetted from a general CSP (X, D, V, S) and an
graph w(H) = (W,F) as described above, then
CSPs represent the same relation on X.
Proof. Assume W = {V1,V2,...,Vw}.
Let an ntuple t a = (~l,Z2,...,zn) be a solution to (X, D, V,
we show the instantiation
of V1,V2,...,Vw,
tb =
(ta[V1],ta[V2],...,ta[Vw]), is a solution to (W, to, F ,/{),
i.e.,
for any (¼,V~) ¯ F,(ta[~],ta[~])¯
RO. If
v~nv~
¢: 0, t~[~][¼
n~]= t~[v~nv~]=t,[r6][¼n v~],
If vks~(l,~, V~) # O, for any V~ ¯ vksa(Vi, V~), (ta[Vi] t~
ta[~])[Vt ] = ta[Vi U Vj][Vt] = ta[V~] ¯ S~. In both
cases, (ta[~],ta[~]) ¯ O.
Let tb = (t~,t2,... ,t~) be a solution to (W, ~, F , R),
we show that t a = (tz t~ tg. t~ ... t~ t~0) is a solution to (X, D, V, S), i.e., for any V~ ¯ V, ta[V~] ¯ S~.
Trivially,
ifV~ ¯ W, ta[V~ ] = t~ ¯ S~. If V~ ¢ W,
then there are ~, V~ ¯ W such that V~ ¯ ~ O V~. So,
t~[v~]= t~[~u V~l[V~]
=(t~[v~]~ t~[~l)[V~]
= (t~
t~)[v~]¯ &.[]
Example 4 Given a CSP as in Example 1, we can
construct an w-graph and transform the given CSP
to a binary CSP as shown in Figure 4 (B). Here,
there are 4 variables {~,V4,V~,Vs}, instead of 10 variables in the dual and join graph representations. Each
variable has a domain which is the same as in the
join graph representation.
There are 3 constraints
on {V2, V4},{V4, Vh},{Vh, V6}, which are more restrictive. For example, pair (4, 4) is in the constraint
{V2, V4}in the join graph representation but is not in
the same constraint in the w-graph representation, because (4, 4) is not allowed by the original constraint
vl = {xl, x2}.
Conclusion
The w-graph is defined as a generalization of the dual
graph in that a dual graph (or join graph) is an w-graph
but an w-graph is not necessarily a dual graph or join
graph. For this reason, the w-graph transformation can
be seen as a generalization of the dual graph transformation, and thus preserves the advantages of the dual
graph transformation when solving a class of general
CSPs where the number of constraints is small and the
constraints are tight.
For CSPs that have large number of constraints and
the constraints have large number of satisfying tuples,
the dual graph transformation is inefficient, mainly because the binary CSP obtained from the dual graph
has large number of variables and large size domains.
As pointed out in Proposition 1, the number of nodes
in an w-graph does not exceed the number of variables
in the original CSP, and therefore, the numberof variables in the w-graph representation has a upper-bound
limit which is independent of the number of original
constraints. Usually, the number of satisfying tuples
varies from constraint to constraint in a given CSP.
The w-cover algorithm given in this paper provides flexibility of choosing those tight constraints to construct
an w-graph, and therefore, the w-graph transformation
avoids the problem of oversized domains.
The w-graph has other nice properties in terms of
cyclicity and width, which have been successfully applied to solving general CSPs directly (Pang 1998).
future work is to apply the w-graph to characterize the
time complexity of solving the binary CSP obtained
from the w-graph transformation and to empirically
compare the efficiency of w-graph based transformation
and w-graph based direct solution approaches.
Mackworth, A. 1977. On reading sketch maps. In
Proceedings of IJCAL 77, 598-606.
Mater, D. 1983. The Theory of Relational Databases.
Computer Science Press.
Pang, W., and Goodwin, S. D. 1996. A new synthesis algorithm for solving CSPs. In Proceedings of
the Pnd International Workshop on Constraint-Based
Reasoning, 1-10.
Pang, W., and Goodwin, S. D. 1997. Constraintdirected backtracking. In The lOth Australian Joint
Conference on AI, 47-56.
Pang, W. 1998. Constraint Structure in Constraint
Satisfaction Problems. Ph.D. Dissertation, University
of Regina, Canada.
Van I-Ientenryck, P. 1989. Constraint Satisfaction in
Logic Programming. Cambridge, MA: The MIT Press.
References
Bacchus, F., and van Beck, P. 1998. On the conversion
between non-binary and binary CSPs. In Proceedings
of AAAI-98, 311-318.
Dechter, R., and Pearl, J. 1989. Tree clustering for
constraint networks. Artificial Intelligence 38:353-366.
Gyssens, M. 1986. On the complexity of join dependencies. ACMTransactions on Database Systems
11(1):81-108.
Jegou, P. 1993a. On some partial line graphs of a hypergraph and the associated matroid. Discrete Mathematics 111:333-344.
Jegou, P. 1993b. On the consistency of general constraint satisfaction problems. In Proceedings of AAAI93, 114-119.
KNOWLEDGE
REPRESENTATION369