1 global constraints: generalised arc consistency it is often important to define n-ary “global”...
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1
Global Constraints: Generalised Arc Consistency
• It is often important to define n-ary “global” constraints, for at least
two reasons
• Ease the modelling of a problem
• Exploitation of specialised algorithms that take the semantics of
the constraint for efficient propagation, achieving generalised arc
consistency.
• The generalised arc consistency (GAC) criterion sees that no value
remains in the domain of a variable with no support in values of each
of the other variables participating in the “global” constraint.
Example: all_diff ([A1, A2, ..., An]
Constrain a set of n variables to be all different among themselves
2
Global Constraints: all_diff
• The constraint definition based on binary difference constraints ()
does not pose any problems as much as modelling is concerned. For
example, it may be being defined recursively in CLP.
• However, constraint propagation based on binary constraints alone
does not provide in general much propagation.
• As seen before, arc consistency is not any better than node
consistency, and higher levels of consistency are in general too
costly and do not take into account the semantics of the all_different
constraint.
one_diff(_,[]).one_diff(X,[H|T]):-
X #\= H, one_diff(X,T).
all_diff([]).all_diff([H|T]):-
one_diff(H,T),all_diff(T).
3
Global Constraints: all_diff
Example:
X1: 1,2,3 X6: 1,2,3,4,5,6,7,8,9
X2: 1,2,3,4,5,6 X7: 1,2,3,4,5,6,7,8,9
X3: 1,2,3,4,5,6,7,8,9 X8: 1,2,3
X4: 1,2,3,4,5,6 X9: 1,2,3,4,5,6
X5: 1,2,3
• It is clear that constraint propagation based on maintenance of node-,
arc- or even path-consistency would not eliminate any redundant
label.
• Yet, it is very easy to infer such elimination with a global view of the
constraint!
4
Global Constraints: all_diff
• Variables X1, X5 and X8 may only take values 1, 2 and 3. Since there
are 3 values for 3 variables, these must be assigned these values
which must then be removed from the domain of the other variables.
• Now, variables X2, X4 and X9 may only take values 4, 5 e 6, that
must be removed from the other variables domains.
X1: 1,2,3 X2: 1,2,3,4,5,6 X3: 1,2,3,4,5,6,7,8,9
X4: 1,2,3,4,5,6 X5: 1,2,3 X6: 1,2,3,4,5,6,7,8,9
X7: 1,2,3,4,5,6,7,8,9 X8: 1,2,3 X9: 1,2,3,4,5,6
X1: 1,2,3 X2: 1,2,3,4,5,6 X3: 1,2,3,4,5,6,7,8,9
X4: 1,2,3,4,5,6 X5: 1,2,3 X6: 1,2,3,4,5,6,7,8,9
X7: 1,2,3,4,5,6,7,8,9 X8: 1,2,3 X9: 1,2,3,4,5,6
5
Global Constraints: all_diff
• In this case, these prunings could be obtained, by maintaining
(strong) 4-consistency.
• For example, analysing variables X1, X2, X5 and X8, it would be
“easy” to verify that from the d4 potential assignments of values to
them, no assignment would include X2=1, X2=2, nor X2=3, thus
leading to the prunning of X2 domain.
• However, such maintenance is usually very expensive,
computationally. For each combination of 4 variables, d4 tuples
whould be checked, with complexity O(d4).
• In fact, in some cases, n-strong consistency would be required, so its
naïf maintenance would be exponential on the number of variables,
exactly what one would like to avoid in search!
6
Global Constraints: all_diff
• However, taking the semantics of this constraint into account, an
algorithm based on quite a different approach allows the prunings to
be made at a much lesser cost, achieving generalised arc
consistency.
• Such algorithm (see [Regi94]), is grounded on graph theory, and
uses the notion of graph matching.
• To begin with, a bipartite graph is associated to an all_diff
constraints. The nodes of the graphs are the variables and all the
values in their domains, and the arcs associate each variable with the
values in its domain.
• In polinomial time, it is possible to eliminate, from the graph, all arcs
that do not correspond to possible assignments of the variables.
7
Global Constraints: all_diff
Key Ideas:
• For each variable-value pair, there is an arc in the bipartite graph.
• A matching, corresponds to a subset of arcs that link some variable
nodes to value nodes, different variables being connected to different
values.
• A maximal matching is a matching that includes all the variable
nodes.
• For any solution of the all_diff constraint there is one and only one
maximal matching.
8
Global Constraints: all_diff
Example: A,B:: 1..2, C:: 1..3, D:: 2..5, E:: 3..6,
all_diff([A,B,C,D,E]).
A
B
C
D
E
1
2
3
4
5
6
A = 1
B = 2
C = 3
D = 4
E = 5
Maximal Matching
9
Global Constraints: all_diff
• The propagation (domain filtering) is done according to the following
principles:
1.If an arc does not belong to any maximal matching, then it does
not belong to any all_diff solution.
2.Once determined some maximal matching, it is possible to
determine whether an arc belongs or not to any maximal matching.
3.This is because, given a maximal matching, an arc belongs to any
maximal matching iff it belongs:
a)To an alternating cycle; or
b)To an even alternating path, starting at a free node.
10
Global Constraints: all_diff
Example: For the maximal matching (MM) shown
• 6 is a free node;
• 6-E-5-D-4 is an even alternating path,
alternating arcs from the MM (E-5, D-4) with
arcs not in the MM (D-5, E-6);
• A-1-B-2-A is an alternating cycle;
• E-3 does not belong to any alternating cycle
• E-3 does not belong to any even alternating
path starting in a free node (6)
• E-3 may be filtered out!
A
B
C
D
E
1
2
3
4
5
6
11
Global Constraints: all_diff
• Compaction
– Before this analysis, the graph may be “compacted”, aggregating,
into a single node, “equivalent nodes”, i.e. those belonging to
alternating cycles.
– Intuitively, for any solution involving these variables and values, a
different solution may be obtained by permutation of the
corresponding assignments.
– Hence, the filtering analysis may be made based on any of these
solutions, hence the set of nodes can be grouped in a single one.
12
Global Constraints: all_diff
• A-1-B-2-A is an alternating cycle;
• By permutation of variables A and B, the
solution <A,B,C,D,E> = <1,2,3,4,5>
becomes <A,B,C,D,E> = <2,1,3,4,5>
• Hence, nodes A e B, as well as nodes 1
and 2 may be grouped together (as may the
nodes D/E and 4/5).
A/B
C
4/5
3
6
D/E
1/2
A
B
C
D
E
1
2
3
4
5
6
With these grouping the graph
becomes much more compact
13
Global Constraints: all_diff
Analysis of the compacted graph shows that
A/B
C
4/5
3
6
D/E
1/2
A/B
C
4/5
3
6
D/E
1/2
• Arc D/E - 3 may be filtered out
(notice that despite belonging to
cycle D/E - 3 - C - 1/2 - D/E, this
cycle is not alternating.
• Arcs D/E - 1/2 and C - 1/2 may
also be filtered.
The compact graph may thus be
further simplified to
14
Global Constraints: all_diff
By expanding back the simplified compact graph, one gets the graph on
the right that
A/B
C
4/5
3
6
D/E
1/2
Which immediately sets C=3 and, more generaly, filters the initial domains to
A,B :: 1..2, C:: 1,2,3, D:: 2,3,4,5, E:: 3,4,5,6
A
B
C
D
E
1
2
3
4
5
6
15
Global Constraints: all_diff
• Upon elimination of some labels (arcs), possibly due to other
constraints, the all_diff constraint propagates such prunings,
incrementally. There are 3 situations to consider:
1. Elimination of a vital arc (the only arc connecting a variable
node with a value node): The constraint cannot be satisfied.
A
B
C
D
E
1
2
3
4
5
6
A
B
C
D
E
1
2
3
4
5
6
?
16
Global Constraints: all_diff
2. Elimination of a non-vital arc which is a member to the maximal
matching
– Determine a new maximal matching and restart from there.
A
B
C
D
E
1
2
3
4
5
6
A
B
C
D
E
1
2
3
4
5
6
Arc A-4 does not belong to the even alternating path started in node 6.
D-5 also leaves this path, but it still belongs to an alternating cycle.
17
Global Constraints: all_diff
3. Elimination of a non-vital arc which is not a member to the maximal
matching
– Eliminate the arcs that do not belong any more to an alternating
cycle or path.
A new maximal matching includes arcs D-5 and E-6. In this matching, arcs
E-4 and E-5 do not belong to even alternating paths or alternating cycles.
A
B
C
D
E
1
2
3
4
5
6
A
B
C
D
E
1
2
3
4
5
6
18
Global Constraints: all_diff
Time Complexity:
Assuming n variables, each of which with d values, and where D is
the cardinality of the union of all domains,
1. It is possible to obtain a maximal matching with an algorithm of
time complexity O(dnn).
2. Arcs that do not belong to any maximal matching may be
removed with time complexity O( dn+n+D).
3. Taking into account these results, we obtain complexity of
O(dn+n+D+dnn). Since D < dn, the total time complexity of the
algorithm is dominated by the last term, thus becoming
O(dnn).
which is much better than the poor result with a naïf analysis.
19
Global Constraints: all_diff
Availability:
1. The all_diff constraint first appeared in the CHIP system (algorithm?).
2. The described implementation is incorporated into the ILOG system,
and avalable as primitive IlcAllDiff.
3. This algorithm is also implemented in SICStus, through buit-in
constraint all_distinct/1.
4. Other versions of the constraint, namely all _different/2, are also
available, possibly using a faster algorithm but with less pruning,
where the 2nd argument controls the available pruning options.
20
Global Constraints: all_diff
Availability:
1. The all_diff constraint first appeared in the CHIP system (algorithm?).
2. The described implementation is incorporated into the ILOG system,
and avalable as primitive IlcAllDiff.
3. This algorithm is also implemented in SICStus, through buit-in
constraint all_distinct/1.
4. Other versions of the constraint, namely all _different/2, are also
available, possibly using a faster algorithm but with less pruning,
where the 2nd argument controls the available pruning options.
21
Global Constraints: all_diff
Example: Sudoku
• Cuts in green are all that are found by maïve all_diff.
• Global all_diff finds all values without backtracking.
• The first cuts are illustrated in the figure.
(where the indices show a possible order in which the cuts are made)
sudoku_cp.pl
Files vh1 e vh2
8 4 6 4 3 5
6 12 37 7 1 8
3 9 8 6
2 1 12 258 6 9 347 5 7
9 4 2 258 6 6 347 5 1
5 258 6 1 347 5 3
6 7 1 1 378 9 2 4
4 2 11 6 378 9 9 3
13 10 8 9 37 7 4 8 5 6 237 10127 10
22
Global Constraints: Assignment
• The all_diff constraint is typically applicable to problems where it is
intended that different tasks are executed by different agents (or use
different resources).
• However, tasks and resources are treated differently. Some are
variables and the others the domains of these variables. For
example, denoting 4 tasks/resources by variablesTi / Rj, one would
have to chose either one of the specifications below
T1 in 1..3, T2 in 2..4, T3 in 1..4, T4 in 1..3.
or
R1 in {1,3,4}, R2 in 1..4, R3 in 1..4, R4 in {2,3}
• Hence, other constraints could be specified either on tasks or on
resources, but not easily involving both types of objects
23
Global Constraints: Assignment
• Such “unfairness” may be overcome by treating both tasks and
resources in a similar way, namely modelling both with distinct
variables.
• These variables still have to adhere to the constraint that different
tasks are performed in different resources.
• Also, if a task j is assigned to resource i, then resource i is assigned
to task j. Denoting tasks by Tj and resources by Ri, the following
condition must stand for any i, j 1..n
Ri = j Tj = i
• This is the goal of global constraint assignment/2, available in
SICStus, that uses the same propagation technique of all_diff.
24
Global Constraints: Assignment
Example:
A1:1..2, A2::1..2, A3::1..3, A4::2..5, A5:: 3..5,
B1:1..3, B2::1..4, B3::3..5, B4::4..5, B5:: 4..5,
assignment([A1,A2,A3,A4,A5], [B1,B2,B3,B4,B5]).
A = 1
B = 2
C = 3
D = 4
E = 5
A1
A2
A3
A4
A5
B1
B2
B3
B4
B5
maximalmatching
25
Global Constraints: Assignment
Since the assignment constraint imposes a maximal matching in the
bipartite graph of tasks Ai and Resources Bj and resources, the same
filtering techniques of the all_diff constraint can be used. Hence the initial
domains are are filtered to
A1:1..2, A2::1..2, A3:: 3 , A4::4..5, A5:: 4..5.
B1:1..2, B2::1..2, B3:: 3 , B4::4..5, B5:: 4..5.
A1
A2
A3
A4
A5
B1
B2
B3
B4
B5
A1
A2
A3
A4
A5
B1
B2
B3
B4
B5
26
Global Constraints: Circuit
• The previous global constraints may be regarded as imposing a
certain “permutation” on the variables.
• In many problems, such permutation is not a sufficient constraint. It is
necessary to impose a certain “ordering” of the variables.
• A typical situation occurs when there is a sequencing of tasks, with
precedences between tasks, possibly with non-adjacency constraints
between some of them.
• In these situations, in addition to the permutation of the variables,
one must ensure that the ordering of the tasks makes a single cycle,
i.e. there must be no sub-cycles.
27
Global Constraints: Circuit
• These problems may be described by means of directed graphs,
whose nodes represent tasks and the directed arcs represent
precedences.
• The arcs may even be labelled by “features” of the precedences,
namely transition times.
• This is a situation typical of several problems of the travelling
salesman type.
A B
C D
A B
C D
2
3
4 5 6
9
8
7
2
2
2
28
Global Constraints: Circuit
• Filtering: For these type of problems, the arcs that do not belong to
any hamiltonian circuit should be eliminated.
• In the graph, it is easy to check that the only possible circuits are
A->B->D->C->A and A->C->D->B->A. Certain arcs (e.g. B->C,
B->B, ...), may not belong to any hamiltonian circuit and can be
safely pruned.
A B
C D
A B
C D
29
Global Constraints: Circuit
• The pruning of the arcs that do not belong to any circuit is the goal of the global constraint circuit/1, available in SICStus.
• This constraint is applicable to a list of domain variables, where the domain of each corresponds to the arcs connecting that variable to other variables, denoted by the order in which they appear in the list.
For example:
A in 2..3, B in 1..4,
C in 1..4, D in 2..3,
circuit([A,B,C,D]).
A/1 B/2
C/3 D/4
A=2
A=3
30
Global Constraints: Circuit
• Global constraint circuit/1 incrementally achieves the pruning of the
arcs not in any hamiltonian circuit. For example, posing
A, D in 2..3, B in 1,2,3,4, C in 1,2,3,4,circuit([A,B,C,D]).
The following prunning is achieved
A in 2..3, B in 1,2,3,4, C in 1,2,3,4, D in 2..3,
since the possible solutions are
[A,B,C,D] = [2,4,3,1] and [A,B,C,D] = [3,1,4,2]
A B
C D
A B
C D
31
Global Constraints: Element
For example, the value X from the arc that
leaves node A, depends on the arc chosen:
if A = 2 then X = 3,
if A = 3 then X = 4;
otherwise X = undefined
A/1 B/2
C/3 D/4
2
3
4 5 6
9
8
7
2
2
2
The disjunction implicit in this definition raises, as well known, problems
of efficiency to constraint propagation.
Often a variable not only has its values constrained by the values of other
variables, but it is actually defined conditionally in function of these values.
32
Global Constraints: Element
In fact, the value of X may only be known upon labelling of variable A.
Until then, a naïf handling of this type of conditoinal constraint would
infer very little from it.
A/1 B/2
C/3 D/4
2
3
4 5 6
9
8
7
2
2
2
However, if other problem constraints impose, for example, X < 4, an
efficient handling of this constraint would impose
not only X = 3 but also A = 2.
if A = 2 then X = 3,
if A = 3 then X = 4;
otherwise X = undefined
33
Global Constraints: Element
• The efficient handling of this type of disjunctions is the goal of global
constraint element/3, available in SICStus and CHIP.
element(X, [V1,V2,...,Vn], V)
• In this constraint, X is a variable with domain 1..n, and both V and
the Vis are either finite domain constraints or constants. The
semantics of the constraint can be expressed as the equivalence
X = i V = Vi
• From a propagation viewpoint, this constraint imposes arc
consistency in X and bounds consistency in V. It is particularly
optimised for when all Vis are ground.
34
Global Constraints: Element & Circuit
• Global constraints may be used together. In particular, constraints
element and circuit may implement the travelling salesman
(satisfaction problem (travelling.pl): For some graph, determine an
hamiltonian circuits whose length does not exceed Max (say 20).
circ([A,B,C,D], Max, Cost):- A in 2..3, B in 1..4, C in {1}\/{3,4}, D in 2..3, circuit([A,B,C,D]), element(A,[_,3,4,_],Ca), element(B,[2,2,5,6],Cb), element(C,[2,_,2,9],Cc), element(D,[_,8,7,_],Cd), Cost #= Ca+Cb+Cc+Cd, Cost #=< Max, labeling([],[A,B,C,D]).
A/1 B/2
C/3 D/4
2
3
4 5 6
9
8
7
2
2
2
35
Global Constraints: Cycle
• Sometimes, the goal is to form more than one (1) circuit connecting
all the nodes.
• This situation corresponds to the classical problem of the Multiple
Traveling Salesman, that is highly suitable to model several
practical applications.
• For example, the case where a number of petrol stations has to be
serviced by a number N of tankers.
• For such type of applications, CHIP extended the global constraint
circuit/1, to another cycle/2, where the first argument is a domain
variable of an integer to account for the number of cycles.
Note: Constraint circuit(L) is implemented in CHIP as cycle(1,L).
36
Global Constraints: Cycle
Example:
X1::[1,3,4], X2::[1,3] , X3::[2,5,6]
X4::[2,5] , X5::[6] , X6::[4,5]
?- N::[1,2,3], X = [X1,X2,X3,X4,X5,X6], cycle(N,X).
N = 1 N = 2 N = 3
X =[3,1,5,2,6,4]X=[3,2,1,5,6,4] X=[3,1,5,2,6,4]
2 4 6
1 3 5
2 4 6
1 3 5
2 4 6
1 3 5
2 4 6
1 3 5
37
Global Constraints: Global Cardinality
• Many scheduling and timetabling problems, have quantitative
requirements of the type
in these N “slots” M must be of type T
• This type of constraints may be formulated with a cardinality
constraint. In some sistems, these cardinality constraints are given
as built-in, or may be implemented through reified constraints.
• In particular, in SICStus the built-in constraint count/4 may be used
to “count” elements in a list, which replaces some uses of the
cardinality constraints (see ahead).
• However, cardinality may be more efficiently propagated if
considered globally.
38
Global Constraints: Global Cardinality
• For example, assume a team of 7 people (nurses) where one or two
must be assigned the morning shift (m), one or two the afternoon
shift (a), one the night shift (n), while the others may be on holliday
(h) or stay in reserve (r).
• To model this problems, let us consider a list L, whose variables Li
corresponding to the 7 people available, may take values in domain
{m, a, n, h, r} (or {1, 2, 3, 4, 5} in languages like SICSTus that require
domains to range over integers).
• Both in SICStus or in CHIP this complex constraint may be
decomposed in several cardinality constraints.
39
Global Constraints: Global Cardinality
SICStus:
count(1,L,#>=,1), count(1,L,#=<,2) % 1 or 2 m/1
count(2,L,#>=,1), count(2,L,#=<,2) % 1 or 2 a/2
count(3,L,#=, 1) , % 1 only
n/3
count(4,L,#>=,0), count(4,L,#=<,2) % 0 to 2 h/4
count(5,L,#>=,0), count(5,L,#=<,2) % 0 to 2 r/5
CHIP:
among([1,2],L,_,[1]), % 1 or 2 m/1
among([1,2],L,_,[2]), % 1 or 2 a/2
among( 1 ,L,_,[3]), % 1 only n/3
among([0,2],L,_,[4]), % 0 to 2 h/4
among([0,2],L,_,[5]), % 0 to 2 r/5
(see [BeCo94] for details):
40
Global Constraints: Global Cardinality
• Nevertheless, the separate, or local, handling of each of these constraints, does not detect all the pruning opportunities for the variables domains. Take the following example:
A,B,C,D::{m,a}, E::{m,a,n}, F::{a,n,h,r}, G ::{n,r}
m (1,2)
F
D
E
A
B
C
G
a (1,2)
n (1,1)
h (0,2)
r (0,2)
41
Global Constraints: Global Cardinality
• A, B, C and D may only take values m and a. Since these may only be attributed to 4 people, no one else, namely E or F, may take these values m and a.
• Since E may now only take value n, that must be taken by a single person, no one else (e.g. F or G) may take value n.
m (1,2)
F
D
E
A
B
C
G
a (1,2)
n (1,1)
h (0,2)
r (0,2)
m (1,2)
F
D
E
A
B
C
G
a(1,2)
n (1,1)
h (0,2)
r (0,2)
42
Global Constraints: Global Cardinality
• This filtering, that could not be found in each constraint alone, can be
obtained with an algorithm that uses analogy with results in
maximum network flows [Regi96].
• A global cardinality constraint gcc/4,
– constrains a list of k variables X = [X1, ..., Xk] ,
– taking values in the domain (with m values) V = [v1, ..., vm],
– such that each of the vi values must be assigned to between Li and Mi variables.
• Then, m SICStus constraints...
count(vi,X,#>=,Li), count(vi,X,#=<,Mi) ...
may be replaced by constraint
gcc([X1....,Xk],[v1,...,vm],[L1,...,Lm],[M1,...,Mm])
43
Global Constraints: Global Cardinality
• The constraint gcc is modelled based on a parallel with a directed graph (or network) with maximum and minimum capacities in the arcs and two additional nodes, a e b. For example:
gcc([A,,...,G],[m,t,n,f,r],[1,1,1,0,0],[2,2,1,2,2])
m (1,2)
F
D
E
A
B
C
G
t (1,2)
n (1,1)
f (0,2)
r (0,2)
1 ; 21 ; 2
1 ; 1
0 ; 20 ; 2
0 ; 1
0 ; 1
0 ;
ab
44
Global Constraints: Global Cardinality
• A solution for the gcc constraint, corresponds to a flow between the
two added nodes, with a unitary flow in the arcs that link variables to
value nodes. In these conditions it is valid the following
Theorem: A gcc constraint with k variables is satisfiable iff there is a
maximal flow of value k, between nodes a and b
i
0
1
i
Flows
m (1,2)
F
D
E
A
B
C
G
t (1,2)
n (1,1)
f (0,2)
r (0,2)
1 ; 21 ; 2
1 ; 1
0 ; 20 ; 2
0 ; 1
0 ; 1
0 ;
2
2
7
b a
45
Global Constraints: Global Cardinality
Of course, being gcc a global constraint it is intended to
1. Obtain a maximum flow with value k, i.e. to show whether the
problem is satisfiable.
2. Achieve generalised arc consistency, by eliminating the arcs
between variables that are not used in any maximum flow solution,
i.e. do not belong to any gcc solution.
3. When some arcs are pruned (by other constraints) redo 1 and 2
incrementally.
In [Regi96] a solution is presented for these problems, together with
a study on its polinomial complexity.
46
Global Constraints: Global Cardinality
1. Obtain a maximal flow of value k
• This optimisation problem may be efficiently solved by linear
programming, that guarantees integer values in the solutions for the
flows.
• However, to take into account the intended incrementality, the
maximal flow may be obtained by using increasing paths in the
residual graph, until no increase is possible.
• The residual graph of some flow f is again a directed graph, with the
same nodes of the initial graph. Its arcs, with lower limit 0, have a
residual capacity that accounts for the non used capacity in a flow f.
47
Global Constraints: Global Cardinality
Residual graph of some flow f
a. Given arc (a,b) with max capacity c(a,b) and flow f(a,b) such that
f(a,b) < c(a,b) there is an arc (a,b) in the residual graph with
residual capacity cr(a,b) = c(a,b) - f(a,b).
The fact that the arc directions are the same means that the flow
may still increase in that direction by up to value cr(a,b).
b. Given arc (a,b) with min capacity l(a,b) and flow f(a,b) such that
f(a,b) > l(a,b) there is an arc (b,a) in the residual graph with residual
capacity cr(b,a) = f(a,b) - l(a,b).
The fact that the arc directions are opposed means that the flow
may decrease the initial flow by up to value cr(b,a).
48
Global Constraints: Global Cardinality
Example:
Given the following flow, with value 6 (lower than the maximal flow, which is of course 7) the following residual graph is obtained
22
2
6
22
1
02
6
2
2
2
49
Global Constraints: Global Cardinality
• Existing an arc (a,b) (in the residual graph) whose flow is not the same
as cr, there might be an overall increase in the flow between arcs a
and b, if the arc belongs to an increasing path of the residual graph.
• In the example below, the path in blue, increases the flow in the
original graph up to its maximum.
22
1
02
7
2
22
2
2
1
2
6
50
Global Constraints: Global Cardinality
• The computation of a maximal flow by this method is guaranteed by the
following
Theorem: A flow f between two nodes is maximal iff there is no
increasing path for f between the nodes.
• A decreasing path between a and b, could be defined similarly as a
path in the residual graph between b and a shown below.
221
02
5
2
122
2
1 6
51
Global Constraints: Global Cardinality
Complexity
• The search for an increasing path may be obtained by a breadth-
first search with complexity O(k+d+), where is the number of
arcs between the nodes and their domains, with size d. As kd
this is the dominant term, and the complexity is O().
• To obtain a maximum flow k, it is required to obtain k increasing
paths, one for each of the k variables. The complexity of the method
is thus O(k).
• As kd, the complexity to obtain a maximal flow k, starting from a
null flow is thus O(k2d). It is of course less, if the starting flow is not
null.
52
Global Constraints: Global Cardinality
• To eliminate arcs that correspond to assignments that do not belong
to any possible solution, we use the following
Theorem:
Let f be a maximum flow between nodes, a and b. For any other
nodes x and y, flow f(x,y) is equal to any flow f´(x,y) induced
between these nodes by another maximal flow f’, iff the arcs (x,y)
and (y,x) are not included in a cycle with more than 3 nodes
that does not include nodes a and b.
• Since the cycles considered do not include both a and b they will not
change the (maximal) flow. Hence, if there are no cycles including
nodes x and y, there are no increasing or decreasing paths through
them that do not change the maximum flow. But then their flow will
remain the same for all maximal flows.
53
Global Constraints: Global Cardinality
• Given the correspondence between maximal flow and the gcc
constraint, if no maximum flow passes in an arc betwen a variable
node X and a value node v, then for no solution of the gcc constraint
is X = v. This can be illustrated in the maximal graph shown below.
22
1
02
7
2
22
2
2
7
2
54
Global Constraints: Global Cardinality
• In the residual graph, the only paths with more than 3 nodes that do
not include nodes a and b, are those shown at the right. Also shown,
in blue, are the arcs with non null flow in the initial situation. These
are all the arcs between variable and value nodes in a maximal flow,
corresponding to possible solutions of constraint gcc/4.
2
2
22
2
7
2
55
Global Constraints: Global Cardinality
• We may compare the initial graph with that obtained after the
elimination of the arcs.
• As expected, the latter fixes value n for variable E, and removes
values m, a and n from variables F and G.
2
m (1,2)
F
D
E
A
B
C
G
a (1,2)
n (1,1)
h (0,2)
r (0,2)
56
Global Constraints: Global Cardinality
Complexity
• Obtaining cycles with more than 3 nodes corresponds to obtain the
subgraphs strongly connected, which can be done in O(m+n) for a
graph with m nodes and n arcs. Here, m = k+d+1 and n kd+d,
hence a global complexity of
O(kd+2d+k+1) O(kd)
• When some constraint removes a value from a variable that was
included in the current maximal flow, a new maximal flow may have
to be recomputed, with complexity at most O(k2d), so the complexity
of using this incremental implementation of the gcc/4 constraint is
O(k2d+kd) O(k2d).
57
Global Constraints: Flow
• Recently [BoPA01] a global constraint was proposed to model and
solve many network flow situations appearing in several problems,
namely transport, communication or production chain applications.
• As the previous ones, the goal of this constraint is to be integrated
with other constraints, as a part of a more general problem, but
allowing the efficient filtering that would not be possible if the
constraint were decomposed into simpler ones.
• Its use is described for problems of maximal flows, production
planning and roastering.
58
Global Constraints: Sequence
• In some applications it is not simply intended to impose cardinality
constraints on a set of variables, i.e. that in this set of variables each
value does not appear more than a certain number of times.
• It is additionally required that these variables are considered in
certain accepted sequences, namely that guarantee that each
value does not appear more than a certain number of times in
every sub-sequence of a given size.
• This situation is well illustrated in the car sequencing problem,
where cardinality constraints must be considered in the context of
some sequencing.
59
Global Constraints: Sequence
Example (Car sequencing):
The goal is to manufacture in an assembly line 10 cars with different
options (1 to 5) shown in the table. Given the assembly conditions of
option i, for each sequence of ni cars, only mi cars can have that
option installed as shown in the table below.
For example, in any sequence of 5 cars, only 2 may have option 4.
instaled.
option capacity 1 2 3 4 5 6 7 8 9 101 1 / 2 X2 2 / 33 1 / 3 X4 2 / 5 X X5 1 / 5
Configuration 1 2 3 4 5 6
cars
X
X
X
X
X
X
XX
60
Global Constraints: Sequence
• Hence, the goal is to implement an efficient global sequence
constraint , gsc/7
gsc(X, V, Seq, Lo, Up, Min, Max)
with the following semantics:
Given a sequence of variables X, Min and Max represent the
minimum/maximum number of times they may take values from list
V. Additionally, in each sequence of Seq variables, Lo and Up
represent the minimum/maximum number of times they may take
values from the list V.
• Notice that in CHIP the gsc/7 constraints has a different syntax
among([Seq, Lo, Up, Min, Max ], X, _, V)
61
Global Constraints: Sequence
• The following program specifies the sequencing constraints of this problem
goal(X):- X = [X1, .. ,X10], L :: 1..6, gcc(X, L, [1,1,2,2,2,2], [1,1,2,2,2,2]),
% gsc(X, V, Seq, Lo, Up, Min, Max) gsc(X, [1,5,6], 2, 1, 1, 5, 5), % option 1
gsc(X, [3,4,6], 3, 2, 2, 6, 6), % option 2 gsc(X, [1,5] , 3, 1, 1, 3, 3), % option 3 gsc(X, [1,2,4], 5, 2, 2, 4, 4), % option 4 gsc(X, [3] , 5, 1, 1, 2, 2). % option 5
option capacity 1 2 3 4 5 6 7 8 9 101 1 / 2 X2 2 / 33 1 / 3 X4 2 / 5 X X5 1 / 5
Configuration 1 2 3 4 5 6
cars
X
X
X
X
X
X
XX
62
Global Constraints: Sequence
• Given the similarity of the constraints, it was proposed in [RePu97]
an efficient implementation of the global sequencing constraint,
gsc/7, based on the global cardinality constraint, gcc/4.
• The way this implementation takes place can be explained by
means of an example.
• Given sequence X = [X1 .. X15], with X :: 1..4, it is intended that
values [1,2] appear between 8 and 12 times, and that each
sequence of 7 variables contains these values between 4 and 5
times , i.e.
gsc(X, [1,2], 7, 4, 5, 8, 12)
63
gsc(X, [1,2], 7, 4, 5, 8, 12)
• We may begin with, by considering a new set of variables Ai,
distributed in subsequences S1, S2 and S3 of 7 elements each
(except the last). The values of each Ak that belong to sequence Sk
are defined as
Xi [1,2] Ai = v , Xi [1,2] Ai = ak
• Clearly, there might not exist less than 2 nor more than 3 values a1
and a2 in the whole sequence A, to guarantee between 4 and 5
values [1,2] in S1 and S2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xi 1 2 1 3 1 2 3 4 2 3 1 1 2 2 4
Ai v v v a1 v v a1 a2 v a2 v v v v a3
Global Constraints: Sequence
64
gsc(X, [1,2], 7, 4, 5, 8, 12)
In sequence S3 there must exist at least 0 appearences of a3 (it can
be completed to the left with sufficient number of vs) and at most 1
instance of a3, to guarantee an adequate number of values [1,2].
In general, for sequences of size S Seq it is necessary to
guarantee that # ak, the number of occurrences of ak to be
max(0, S - Up) # ak Seq - Lo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xi 1 2 1 3 1 2 3 4 2 3 1 1 2 2 4
Ai v v v a1 v v a1 a2 v a2 v v v v a3
Global Constraints: Sequence
65
gsc(X, [1,2], 7 , 4, 5, 8, 12)
gsc(X, V , Seq , Lo, Up, Max, Min)
• One must still guarantee the existence of between 8 (Min) and 12
(Max) vs in the Ai sequences.
• All these conditions are met, with the constraints that map the Xis
into Ais, and also with the global cardinality constraint
gcc(A, [v,a1,a2,a3], [8,2,2,0], [12,3,3,1])
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xi 1 2 1 3 1 2 3 4 2 3 1 1 2 2 4
Ai v v v a1 v v a1 a2 v a2 v v v v a3
Global Constraints: Sequence
66
• Not all sequences of 7 (Seq) elements were already considered. For
thus purpose, 7 (Seq) gcc constraints must be considerd as shown
in the figure
• All these conditions are met, with the constraints that map the Xis
into Ais, and also with the global cardinality constraint
gcc(A, [v,a1,a2,a3], [8,2,2,0], [12,3,3,1])
Global Constraints: Sequence
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xi 1 2 1 3 1 2 3 4 2 3 1 1 2 2 4
Ai a1 a1 a2 a2 a3
Bi b2 b2 b2 b3 b3
Ci c2 c2 c2 c3 c3
Di d2 d2 d2 d2 d3
Ei e1 e2 e2 e2 e3
Fi f1 f2 f2 f2 f3
Gi g1 g2 g2 g2 g3
67
The gsc constraint is thus mapped into 7 gcc constraints
gcc(A, [v,a1,a2,a3], [8,2,2,0], [12,3,3,1])gcc(B, [v,b1,b2,b3], [8,0,2,2], [12,1,3,3])gcc(C, [v,c1,c2,c3], [8,0,2,1], [12,2,3,3])gcc(D, [v,d1,d2,d3], [8,0,2,2], [12,3,3,3])gcc(E, [v,e1,e2,e3], [8,0,2,0], [12,3,3,3])gcc(F, [v,f1,f2,f3], [8,0,2,0], [12,3,3,3])gcc(G, [v,g1,g2,g3], [8,1,2,0], [12,3,3,2])
Global Constraints: Sequence1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xi 1 2 1 3 1 2 3 4 2 3 1 1 2 2 4
Ai a1 a1 a2 a2 a3
Bi b2 b2 b2 b3 b3
Ci c2 c2 c2 c3 c3
Di d2 d2 d2 d2 d3
Ei e1 e2 e2 e2 e3
Fi f1 f2 f2 f2 f3
Gi g1 g2 g2 g2 g3
68
Global Constraints: Stretch
• Recently ([Pesa01]), another global constraint was proposed to
handle globally situations where it is intended to “stretch” the
sequences of equal values.
• Its use has been shown in roastering problems (nurse timetables).
For example, to guarantee that a nurse has a maximum sequence
of day shifts, not a sequence of alternating day and night shifts.
• Again, this constraint is to be integrated with other constraints to be
part of a more general problem, but allowing the efficient filtering
that would not be possible if the constraint were decomposed into
simpler ones.
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