We describe a set of necessary conditions that are useful for generating propagation algorithms for the cardinality operator as well as for over-constrained problems with preferences. Constructive disjunction as well as the entailments rules originally proposed for the cardinality operator can be seen as simple cases of these necessary conditions. In addition these necessary conditions have the advantage of providing more pruning

Beldiceanu, Nicolas

Carlsson, Mats

English

RISE – Research Institutes of Sweden

Constructive Cardinality

Swedish Institute of Computer Science Publications Database

Constructive Cardinality 
Nicolas Beldiceanu  and  Mats Carlsson 
S I C S , Lägerhyddsvägen 18, SE-75237 Uppsala, Sweden 
{nicolas,matsc}@sics.se 
 
 
August 14 2001 
SICS Technical Report T2001:15 
ISSN 1100-3154 
ISRN: SICS-T--2001:15-SE 
 
Abstract. We describe a set of necessary conditions that are useful for 
generating propagation algorithms for the cardinality operator as well as for 
over-constrained problems with preferences. Constructive disjunction as well as 
the entailments rules originally proposed for the cardinality operator can be 
seen as simple cases of these necessary conditions. In addition these necessary 
conditions have the advantage of providing more pruning. 
Keywords: Combinatorial problems, cardinality constraint, constructive 
disjunction, soft constraints. 
 
CONSTRUCTIVE CARDINALITY 
Nicolas BELDICEANU  and  Mats CARLSSON 
S I C S , Lägerhyddsvägen 18, SE-75237 Uppsala, Sweden 
{nicolas,matsc}@sics.se 
We describe a set of necessary conditions that are useful for generating propagation algorithms for the 
cardinality operator as well as for over-constrained problems with preferences. Constructive disjunction as 
well as the entailments rules originally proposed for the cardinality operator can be seen as simple cases of 
these necessary conditions. In addition these necessary conditions have the advantage of providing more 
pruning. 
Keywords: Combinatorial problems, cardinality constraint, constructive disjunction, soft constraints 
1  Introduction 
For definitions from constraint programming not given here we refer to [3]. A domain 
variable V is a variable that ranges over a finite set of integers; ( )Vdom , ( )Vmin  and 
( )Vmax  respectively denote the set of possible values, the minimum and the 
maximum values of V. A constraint ( )nVVCTR ,..,1  over an ordered set of variables 
nVV ,..,1  is a relation over the cartesian product of the domains of nVV ,..,1 , while the 
negation ( )nVVCTR ,..,1¬  is the complement with respect to that cartesian product. To 
any constraint or conjunction of constraints CT , we associate a 0-1 domain variable 
CTB  that is equal to 0 if CT  does not hold, and 1 if CT  holds. We note 
( )CTBCT max# = . 
Since its introduction in 1991 within constraint programming, the cardinality 
operator [2] has been recognized as a generic combinator which was progressively 
integrated into several constraint systems. Its most general form is ( ) ( ){ }( )
nnknnk VVCTRVVCTRC ,..,,..,,..,,ycardinalit 11111 1  where C  is a domain variable and ( ) ( ){ }
nnknnk VVCTRVVCTR ,..,,..,,.., 11111 1  is a collection of constraints over finite domain 
variables. The cardinality operator holds iff ( )∑
=
#=
n
i
ikii iVVCTRC
1
1,..,
1
. Throughout this 
paper we consider a more general case of the cardinality operator where we associate 
to each constraint iCTR ( )ni ≤≤1  a weight IN∈iW ; the weighted cardinality operator 
                                                           
1
 As usual within constraint programming, this definition applies for the ground case when all 
the variables of the constraints nCTRCTR ,..,1  are fixed. 
holds iff ( )( )∑
=
#⋅=
n
i
ikii iVVCTRC iW
1
1,.., . This extension is for instance useful for those 
problems where the constraints have different weights and where the objective is to 
maximize the sum of the weights of the constraints that hold. 
From an operational point of view the cardinality operator used entailment [4] in 
order to implement the corresponding propagation. However a fundamental weakness 
is that it assumes each constraint to be independent. This is not the case in practice 
since, very often, the same variable occurs in more than one constraint. The 
contribution of this paper is to propose new necessary conditions that take advantage 
of the fact that some variables occur in more than one constraint. In addition it also 
discusses possible ways to use them for pruning variables. 
2  Necessary Conditions for the weighted cardinality Operator 
Consider any partitioning of the collection of constraints ( ) ( ){ }
nnknnk VVCTRVVCTR ,..,,..,,.., 11111 1  of the weighted cardinality operator into 
p ( )np ≤≤1  conjunctions of constraints, where iC ( )pi ≤≤1  denotes the set of indices 
of those constraints that belong to the i th conjunction, and iC  the number of 
elements of iC . 
Theorem 1. The weighted cardinality operator can be satisfied only if :   
      
( ) ∑
= ⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∈
#−⋅
∈ ⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎠⎞⎜⎝⎛−≤ ∧∑
=
p
i jjk
VjVjCTR
iCj
jW
iCj
n
i
iWC
1
,..,11min
1
min   .       (1) 
Proof. For each conjunction of constraints such that we find a contradiction, we have 
that at least one constraint can’t be satisfied, so we compute the minimum weight of 
the constraints of that conjunction. Since no constraint occurs in more than one 
conjunction, and by adding the previous minimum weights, we get a lower bound of 
the sum of the weight of constraints that do not hold. By subtracting it from the sum 
of the weights of all constraints, we obtain an upper bound of the sum of the weights 
of the constraints that hold. Since C  should not be greater than this upper bound, a 
necessary condition is as follows: the minimum value of C  should be less than or 
equal to this upper bound.              ❏ 
Theorem 1 allows adjusting the maximum value of C . The conjunctions of 
constraints can be obtained e.g. by using the following heuristic which tries to extend 
an existing conjunction as long as contradiction is not reached: in order to reach 
contradiction more rapidly the next constraint to incorporate into a conjunction should 
share as many variables as possible with the variables that already occur in the 
constraints of that conjunction. 
As a corollary of Theorem 1 we have Theorem 2 which allows enforcing a 
conjunction of constraints. 
Theorem 2. The weighted cardinality constraint can be satisfied only if each 
conjunction of constraints ( )phh ≤≤1 , such that condition (2) is true, holds. 
( ) ( ) ∑
≠= ⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∈
#−⋅
∈ ⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎠⎞⎜⎝⎛−−> ∧∈
=
∑
p
hii jjk
VjVjCTR
iCj
jW
iCj
i
Ci
n
i
i WWC
h
,1
,..,11minminmin
1
 (2) 
Theorem 2 can be interpreted as follows: if C  is for sure greater than the upper 
bound obtained by considering all constraints except the one that corresponds to the 
minimum weight in conjunction h , then each constraint of conjunction h  should be 
enforced. 
Finally we explain how to prune the domain of a variable V  which occurs in at 
least one constraint of the weighted cardinality operator. 
Theorem 3. The weighted cardinality operator can be satisfied only if condition 
(3) or (4) holds, for every V  and ( )Vval dom∈ . 
  valV ≠               (3) 
  
( ) ∑
= ⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
=∧⎟⎟⎠
⎞
⎜⎜⎝
⎛
∈
#−⋅
∈ ⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎠⎞⎜⎝⎛−≤ ∧∑
=
p
i
valVjjkVjVjCTR
iCj
jW
iCj
n
i
iWC
1
,..,11min
1
min  (4) 
Proof. The right hand side of condition (4) corresponds to an upper bound of the sum 
of the weights of the constraints that hold for any conjunction of constraints where 
valV = . If C  can’t be less than or equal to this upper bound, then the only way to 
avoid violating the weighted cardinality operator is to remove value val  from 
( )Vdom .                ❏ 
A first possible heuristic for using Theorem 3 for pruning a variable V  is as 
follows: partition the constraints occurring in the weighted cardinality operator in 
such a way that each constraint mentioning variable V  is put in a singleton 
conjunction. Let denote VmV CTRCTR ,..,1  the constraints where V  occurs and let ( )iCTRV⏐dom  ( )ni ≤≤1  denote the domain of variable V  under the assumption that 
constraint iCTR  is enforced (it is empty if constraint iCTR  leads to a contradiction). 
Furthermore let d  be the sum of the weights of those constraints iCTR { }( )VmVi ,..,1∈ . 
Thus, from Theorem 3 we derive the following rule: If 
( ) ( ) ( )
{ }
d
VmVC
pi
jkjj
Cj
jCj
n
i
i
i
j
ii
VVCTRWWC −∑ ∧∑
∅=∩
∈ ∈∈= ⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
#−⋅−>
,..,1
,..1
1
1
,..,1minmin  then all 
values ( )
{ }

VmVi
iCTRVval
,..,1
dom
∈
⏐∉  should be removed from ( )Vdom . 
Theorem 3 can also be used for adjusting the minimum value of a variable V  
occurring in the cardinality operator (i.e. we assume all weights to be equal to 1). Let ( )CTV⏐min  denotes the minimum of variable V  under the assumption that the 
constraint or the conjunction of constraints CT  is enforced (it is equal to ( ) 1max +V  if 
CT  leads to a contradiction). Furthermore, let us define the thk  smallest value of a 
multiset of natural numbers lXX ,..,1  ( )kl ≥  as the smallest value iX  ( )li ≤≤1  such 
that there exist at least 1−k  jX  ( )ljij ≤≤≠ 1,  that are less than or equal to iX . 
Again from Theorem 3 we get the following pruning rule: 
If ( ) pnC −>min  then we can adjust the minimum value of V  to the 
( )( )pnC +−min th smallest value among values ( )⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⏐∧
∈
j
i
jkjj
Cj
VVCTRV ,..,min 1  ( )pi ≤≤1 . 
We have a similar rule for adjusting the maximum value of variable V . 
 
Three theorems similar to Theorems 1, 2, 3 can also be obtained by considering 
conjunctions of negation of constraints. The pruning rules of the cardinality operator 
provided in [2] can be derived from Theorems 1 and 2 by setting the number of 
constraints in a conjunction, as well as all the weights, to 1. Constructive disjunction 
[1], [4] can also be obtained from Theorem 3 by associating each alternative of the 
disjunction to a singleton conjunction, by setting C  to 1, and by setting all the 
weights to 1. For a specific constraint network, how to find out the way to partition 
the constraints of the weighted cardinality operator that leads to the maximum amount 
of pruning, is a natural question that arises from this paper. 
Acknowledgments 
Thanks to Per Kreuger, Per Mildner and Emmanuel Poder for helpful comments and 
suggestions on this paper. 
References 
1. Würtz, J., Müller, T.: Constructive Disjunction Revisited. In 20th German Annual 
Conference on Artificial Intelligence. LNAI vol. 1137, 377-386, Springer-Verlag, (1996). 
2. Van Hentenryck, P., Deville, Y.: The Cardinality Operator: A New Logical Connective for 
Constraint Logic Programming. In International Conference on Logic Programming. The 
MIT Press, 745-759, (1991). 
3. Van Hentenryck, P.: Constraint Satisfaction in Logic Programming. The MIT Press, (1989). 
4. Van Hentenryck, P., Saraswat, V., Deville, Y.: Design, Implementation and Evaluation of 
the Constraint Language cc(FD). In A. Podelski, ed., Constraints: Basics and Trends, vol. 
910 of Lecture Notes in Computer Science, Springer-Verlag, (1995). 
 


Constraint Satisfaction in Logic Programming.

Constructive Disjunction Revisited.

Implementation and Evaluation of the Constraint Language cc(FD).

The Cardinality Operator: A New Logical Connective for Constraint Logic Programming.

Digitala Vetenskapliga Arkivet - Academic Archive On-line

http://soda.swedish-ict.se/2389/1/SICS-T--2001-15--SE.ps.Z

Constructive Cardinality

Abstract

Similar works

Full text

Available Versions

RISE – Research Institutes of Sweden

Swedish Institute of Computer Science Publications Database

Digitala Vetenskapliga Arkivet - Academic Archive On-line