Non-binary constraints Toby Walsh Cork Constraint Computation Center [email protected].

Non-binary constraints

Toby Walsh

Cork Constraint Computation Center

[email protected]

Global constraints

Toby Walsh

Cork Constraint Computation Center

[email protected]

Non-exhaustive catalog

Order constraints Constraints on values Partitioning constraints Timetabling constraints Graph constraints Scheduling constraints Bin-packing constraints

Global constraints

It isn’t just all-different!

Many constraints specialized to application domains Scheduling Packing ..

Order constraints

min(X,[Y1,..,Yn]) and max(X,[Y1,..Yn])X <= minimum(Y1,..,Yn)X >= maximum(Y1,..Yn)n.b. min is relational not functional

Order constraints

min(X,[Y1,..,Yn]) and max(X,[Y1,..Yn])

X <= minimum(Y1,..,Yn)

X <= maximum(Y1,..Yn)

n.b. min is relational not functional

min_mod(X,[Y1,..,Yn],m) and max_mod(X,[Y1,..Yn],m)

X mod m <= minimum(Y1 mod m,..,Yn mod m)

X mod m <= maximum(Y1 mod m,..,Yn mod m)

Order constraints

min_n(X,n,[Y1,..Ym]) and max_n(X,n,[Y1,..,Ym)

X is nth smallest value in Y1,..YmX is nth largest value in Y1,..Ym

Order constraints

min_n(X,n,[Y1,..Ym]) and max_n(X,n,[Y1,..,Ym)

X is nth smallest value in Y1,..YmX is nth largest value in Y1,..Ym

min_n(X,1,[Y1,..,Ym]) => min(X,[Y1,..,Yn])max_n(X,1,[Y1,..,Ym]) => max(X,[Y1,..,Ym])

n.b. min_n is functional but min is relational (any X smaller than min will do!

Value constraints

among(N,[Y1,..,Yn],[val1,..,valm])N vars in [Y1,..,Yn] take values val1,..valme.g. among(2,[1,2,1,3,1,5],[3,4,5])

Value constraints

among(N,[Y1,..,Yn],[val1,..,valm])

N vars in [Y1,..,Yn] take values val1,..valm

e.g. among(2,[1,2,1,3,1,5],[3,4,5])

count(n,[Y1,..,Ym],op,X) where op is =,<,>,=/,<= or >=

relation “Yi op X” holds n times

E.g. among(n,[Y1,..,Ym],[k]) = count(n,[Y1,..,Ym],=,k)

Value constraints

balance(N,[Y1,..,Yn])

N = occurrence of more frequent value - occurrence of least frequent value

E.g balance(2,[1,1,1,3,4,2])

Value constraints

balance(N,[Y1,..,Yn])

N = occurrence of more frequent value - occurrence of least frequent value

E.g balance(2,[1,1,1,3,4,2])

all-different([Y1,..,Yn]) => balance(0,[Y1,..,Yn])

Value constraints

min_nvalue(N,[Y1,..,Yn]) and max_nvalue(N,[Y1,..,Yn])

least (most) common value in Y1,..,Yn occurs N times

E.g. min_nvalue(2,[1,1,2,2,2,3,3,5,5])

Can replace multiple count or among constraints

Value constraints

common(X,Y,[X1,..,Xn],[Y1,..,Ym])

X vars in Xi take a value in Yi

Y vars in Yi take a value in Xi

E.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9])

Value constraints

common(X,Y,[X1,..,Xn],[Y1,..,Ym])

X vars in Xi take a value in Yi

Y vars in Yi take a value in Xi

E.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9])

among(X,[Y1,..,Yn],[val1,..,valm]) = common(X,Y,[X1,..,Yn],[val1,..,valm])

Value constraints

same([X1,..,Xn],[Y1,..,Yn])Yi is a permutation of Xi

Value constraints


used_by([X1,..,Xn],[Y1,..,Ym])all values in Yi are used by vars in Xim>=m

Value constraints


used_by([X1,..,Xn],[Y1,..,Ym])all values in Yi are used by vars in Xim>=m

on n values:alldifferent([X1,..,Xn])=same([X1,..,Xn],[1,..,n])

=used_by([X1,..,Xn],[1,..,n])

Partitioning constraints

all-different([X1,..,Xn])



Other flavoursall-different_except_0([X1,..,Xn])Xi=/Xj unless Xi=Xj=0

0 is often used for modelling purposes as “dummy” value Don’t use this slab Don’t open this bin ..



Other flavourssymmetric-all-different([X1,..,Xn])Xi=/Xj and Xi=j iff Xj=i

Very common in practice Team i plays j iff Team j plays i..



Other flavourssymmetric-all-different([X1,..,Xn])Xi=/Xj and Xi=j iff Xj=i

Very common in practice Team i plays j iff Team j plays i.. Regin has proposed very efficient algorithm


nvalue(N,[X1,..,Xn])

Xi takes N different values

all-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn)


nvalue(N,[X1,..,Xn])

Xi takes N different values

all-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn)

gcc([X1,..,Xn],Lo,Hi)

values in Xi occur between Lo and Hi times

all-different([X1,..,Xn])=gcc([X1,..,Xn],1,1)

Timetabling constraints

change(N,[X1,..,Xn]),op) where op is {=,<,>,<=,>=,/=}

“Xi op Xi+1” holds N times

E.g. change(3,[4,4,3,4,1],/=)

You may wish to limit the number of changes of classroom, shifts, …

Timetabling constraints

longest_changes(N,[X1,..,Xn]),op) where op is {=,<,>,<=,>=,/=}

longest sequence “Xi op Xi+1” is of length N

E.g. longest_changes(2,[4,4,4,3,3,2,4,1,1,1],=)

You may wish to limit the length of a shift without break, …

Graph constraints

Tours in graph a often represented by the successors:

[X1,..,Xn] means from node i we go to node Xi

Graph constraints

Tours in graph a often represented by the successors:

[X1,..,Xn] means from node i we go to node Xi E.g. [2,1,5,3,4] represents the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3)

Graph constraints

cycle(N,[X1,..,Xn])

there are N cycles in Xi

e.g. cycle(2,[2,1,5,3,4]) as we have the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3)

Useful for TSP like problems (e.g. sending engineers out to repair phones)

Graph constraints

derangement(N,[X1,..,Xn])

there are no length 1 cycles in Xi

Graph constraints



e.g. derangement([2,1,5,3,4]) as the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3) have length 2

and 3

Graph constraints



e.g. derangement([2,1,5,3,4]) as the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3) have length 2

and 3

e.g. derangement([3,1,2,4]) as the 2 cycles (1)->(3)->(2)->(1) and (4)->(4) are of length 3 and 1

Graph constraints

We may also wish to represent predecessors For good labelling heuristics For ease of statement of constraints

[X1,..,Xn] means we came to node i from node Xi

Graph constraints


[X1,..,Xn] means we came to node i from node Xi E.g. [2,1,4,5,3] now represents the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3)

Graph constraints



(1)->(2)->(1) and (3)->(5)->(4)->(3) In successor notation this was [2,1,5,3,4]

Graph constraints




How do we know they are the same?

Graph constraints




How do we know they are the same? With a channelling constraint!

Graph constraints

Inverse([X1,..,Xn],[Y1,..,Yn])

Where [X1,..,Xn] is a successor representation of a path, and [Y1,..,Yn] is a predecessor

Thus inverse([2,1,4,5,3],[2,1,5,3,4])

Graph constraints

Inverse([X1,..,Xn],[Y1,..,Yn])

Where [X1,..,Xn] is a successor representation of a path, and [Y1,..,Yn] is a predecessor

Thus inverse([2,1,4,5,3],[2,1,5,3,4])

And inverse([2,1,5,3,4],[2,1,4,5,3])

Scheduling constraints

cummulative([S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],L)

schedules n (concurrent) jobs, each with a height Hi

ith job starts at Si, runs for Di and ends at Ei Ei=Si+Di

at any time, accumulated height of running jobs is less than L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25


coloured_cummulative([S1,..,Sn],[D1,..,Dn],[E1,..,En],[C1,..,Cn],L)

schedules n (concurrent) each with a colour Ci no more than L colours running at any one time

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25


cycle_cummulative(m,[S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],L)

schedules n (concurrent) jobs, each with a height Hi onto a cyclic schedule of length m


cummulatives([M1,,,Mn],[S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],[L1,..,Lm])

schedules n (concurrent) jobs, each with a height Hi onto one of m machines

ith runs on Mi accumulated height of running jobs on machine i

<= Li


cummulatives([M1,..Mn],[S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],[L1,..,Lm])

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Machine 1

Machine 2


coloured_cummulatives([M1,,,Mn],[S1,..,Sn],[D1,..,Dn],[E1,..,En],[C1,..,Cn],[L1,..,Lm])

schedules n (concurrent) jobs, each with a colour i onto one of m machines

ith runs on Mi number of colours of running jobs on machine i

<= Li

Bin-packing constraints

bin_packing(capacity,[B1,..,Bn],[w1,..,wn])

for each bin j, sum_Bi=j wi <= capacity




special case of cummulative with task durations=1




special case of cummulative with task durations=1

unlike normal bin-packing, does not minimize the number of open bins




special case of cummulative with task durations=1!


how could we do this?




special case of cummulative with task durations=1!


Bi=j implies Open_j=1

Minimize sum_j Open_j

Misc constraints

element(Index,[a1,..,an],Var)

Var=a_Index

constraint programming’s answer to arrays!

e.g. element(Item,[10,23,12,15],Cost)

GAC algorithms

GAC-Schema

A generic framework for achieving AC for any kind of constraint (can be non binary).Bessiere and Regin, IJCAI’97

You just have to say how to compute a solution. Works incrementality (notion of support).

GAC-Schema: instantiation

List of allowed tuples List of forbidden tuples Predicates Any OR algorithm Solver re-entrace

GAC-Schema

Idea: tuple = solution of the constraintsupport = valid tuple- while the tuple is valid: do nothing

- if the tuple is no longer valid, then search for a new support for the values it contains

a solution (support) can be computed by any algorithm

Example

X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)}

Example

X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)} Support for (x1,a): (a,a,a) is computed and (a,a,a) is added to S(x2,a) and

S(x3,a), (x1,a) in (a,a,a) is marked as supported.

Example


S(x3,a), (x1,a) in (a,a,a) is marked as supported. Support for (x2,a): (a,a,a) is in S(x2,a) it is valid, therefore it is a support.

(Multidirectionnality). No need to compute a solution

Example


S(x3,a), (x1,a) in (a,a,a) is marked as supported. Value a is removed from x1, then all the tuples in S(x1,a) are no longer valid:

(a,a,a) for instance. The validity of the values supported by this tuple must be reconsidered.

Example


S(x3,a), (x1,a) in (a,a,a) is marked as supported. Support for (x1,b): (b,b,a) is computed, and updated ...

GAC-Schema: complexity

CC complexity to check consistency (seek in table, call to OR algorithm): seek for a Support costs CC

n variables, d values:for each value: CCfor all values: O(ndCC)

For any OR algorithm which is able to compute a solution, Arc consistency can be achieved in O(ndCC).

Table Constraint: An example Configuration problem:

5 types of components: {glass, plastic, steel, wood, copper}3 types of bins: {red, blue, green} whose capacity is red 5, blue 5, green 6Constraints:- red can contain glass, cooper, wood- blue can contain glass, steel, cooper- green can contain plastic, copper, wood- wood require plastic; glass exclusive copper- red contains at most 1 of wood- green contains at most 2 of woodFor all the bins there is either no plastic or at least 2 plasticGiven an initial supply of 12 of glass, 10 of plastic, 8 of steel, 12 of wood and 8 of copper; what is the minimum total number of bins?

Table Constraint: results

#bk timestandard model 1,361,709 430Table Constraint 12,659 9.7

Semantics of a constraint

Speed-up the search for a support (solution which contain a value (x,a))

Exploiting constraint semantics

Speed-up the search for a support (solution which contain a value (x,a)):x < y, D(x)=[0..10000], D(y)=[0..10000]support for (x,9000): immediate any value greater than 9000 in D(y)

Semantics of a constraint

Design of ad-hoc filtering algorithm:x < y : (a) max(x) = max(y) -1(b) min(y) = min(x) +1

Exploiting semantics

Design of ad-hoc filtering algorithm:x < y : (a) max(x) = max(y) -1(b) min(y) = min(x) +1

Triggering of the filtering algorithm:no possible pruning of D(x) while max(y) is not modified

no possible pruning of D(y) while min(x) is not modified

Regin’s alldiff algorithm

Construct the value graph

x1

x2

x3

x4

x5

x6

1

2

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7

The value graph:

D(x1)={1,2}D(x2)={2,3}D(x3)={1,3}D(x4)={3,4}D(x5)={2,4,5,6}D(x6)={5,6,7}

Alldiff constraint

Construct a flow network

Flow network

x1

x2

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x6

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s

Default orientation

Flow network

x1

x2

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x6

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t s

Default orientation

Flow network

x1

x2

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x6

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t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

Alldiff constraint

Construct a flow network Compute a feasible flow

Flow network

x1

x2

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x6

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t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

A feasible flow

x1

x2

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x6

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t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

Alldiff constraint

Construct a flow network Compute a feasible flow Construct the residual graph

Reverse direction of arcs on feasible flow (that is, of red arcs)

A feasible flow

x1

x2

x3

x4

x5

x6

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t s

(6,6)

(0,1)

(0,1)

(1,1)

Default orientation

Residual graph

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x6

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orientation

Alldiff constraint

Construct a flow network Compute a feasible flow Construct the residual graph Compute the strongly connected components

Strongly connected component = maximal subgraph such that if directed path u to v, also from v to u

Alldiff constraint


Strongly connected component = maximal subgraph such that if directed path u to v, also from v to u

Exists O(N) algorithm due to Tarjan (1972) = O(n+d)

Residual graph

x1

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orientation

Alldiff constraint


Remove arcs of flow value 0 (i.e. black arcs) with ends belonging to two different components

Residual graph

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orientation

GAC for all-different

TheoremAn arc is deleted from residual graph iff the associated

value is not GAC

Residual graph

x1

x2

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x6

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orientation

GAC subproblem

x1

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The value graph:

D(x1)={1,2}D(x2)={2,3}D(x3)={1,3}D(x4)={4}D(x5)={5,6}D(x6)={5,6,7}

Before!

x1

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The value graph:

D(x1)={1,2}D(x2)={2,3}D(x3)={1,3}D(x4)={3,4}D(x5)={2,4,5,6}D(x6)={5,6,7}

Conclusions

Vast range of global constraints Cover all the areas that CP cover Some very generic (e.g. all-different) Some very specific (coloured cummulative ..)

GAC algorithms Generic (GAC-schema) Specific (GAC alldifferent)

Non-binary constraints Toby Walsh Cork Constraint Computation Center [email protected].

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Transcript of Non-binary constraints Toby Walsh Cork Constraint Computation Center [email protected].