Global Constraints (continued) Nicolas Beldiceanu SICS - - PowerPoint PPT Presentation

Global Constraints (continued)

Nicolas Beldiceanu SICS

Lägerhyddsvägen 5 SE-75237 Uppsala, Sweden (nicolas@sics.se)

Except resource scheduling, routing, geometric, regulation and optimisation constraints (see in next courses). But first, go into more detail on how to describe global constraints.

Describe other global constraints

V1 V2 V3 V4 V5 Edge constraint:

=

V1 V2 V3 V4 V5 = = = = = = = = = = = = = = =

3 1 7 1 6

= = = = = = Graph property:

NSCC=NVAL

nvalue(4, { var-3,var-1,var-7,var-1,var-6 })

Principle of description

Arc generators

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

SELF LOOP PATH PATH PATH_1 PATH_N CIRCUIT CHAIN CYCLE CLIQUE CLIQUE(≠ ≠ ≠ ≠) CLIQUE(<) PRODUCT PRODUCT(=) SYMMETRIC_PRODUCT SYMMETRIC_PRODUCT(=)

V1 V2 V3 V4 V5 Edge constraint:

=

V1 V2 V3 V4 V5 = = = = = = = = = = = = = = =

3 1 7 1 6

= = = = = = Graph property:

NSCC=NVAL

nvalue(4, { var-3,var-1,var-7,var-1,var-6 })

More details

Derived collections Set variables More than one graph Constraints on subsets of vertices

Motivations for Set Variables (finite set of integers)

1) Exists since a long time (both in academy and in industrial solvers) 2) Has a discrete nature (as standard domain variables) 3) Expressive (avoid models with artificial variables) 4) Recently some suggestions of global constraints with set variables 5) Within linear programming and constraint programming a typical constraint is: for a given graph selects a subset of arcs so that a given graph property holds

Inserting Set Variables Has to introduce: 1) A new basic data type: finite set of integers 2) Set variables 3) Elementary constraints over set variables

Typical Arc Constraint

index1: int var1 : svar index2∈var1 index2: int var2 : svar index1: 1 var1: {2,3,5} 3∈var1 index3: 3 var3 : {} index2: 2 var2 : {} index4: 4 var4 : {} 2∈var1 4∈var1

Typical Arc Constraint

index1: int var1 : svar index2∈var1 index2: int var2 : svar index1: 1 var1: {2,3,5} 3∈var1 index3: 3 var3 : {} index2: 2 var2 : {} index4: 4 var4 : {} 2∈var1 4∈var1

Example of Global Constraint with Set Variables Given a directed graph select a subset of vertices so that the corresponding sub-graph does not contain any circuit. (F.Fages)

Example of Global Constraint with Set Variables

Symbolic Constraints in Integeger Linear Programming

[E.Althaus,A.Bockmayr,M.Elf,M.Jünger,T.Kasper,K.Mehlhorn]

http://www.mpi-sb.mpg.de/SCIL/

StronglyConnected

”This symbolic constraints takes as arguments a directed graph G and a var_map<edge> X. X has to map every edge of the graph to a binary variable. The feasible assignments of the symbolic constraint are those where the vector of the variables associated with the edges of the graph is an incidence vector of a strongly connected subgraph of G.”

Constraints Over Several Final Graphs

PROBLEM

Want to check graph properties on several final graphs

EXAMPLES

• global cardinality [RÉGIN]: impose restriction on number of occurrences
• f a value (depend on the value).
• stretch[PESANT]: impose minimum and maximum time that a value can
• ccur in a consecutive way (depend on the value)

SOLUTION Introduce an iterator over the items of a collection for

specifying in a generic way a set of:

• elementary constraints which are pairwise incompatible (before CTR & ¬CTR)
• graph properties.

Constraints Over Set of Vertices

CONTEXT

Want to express constraints on set of variables which are not yet defined.

EXAMPLES

• impose a constraint on all the objects which are assigned to the same bin.
• impose a constraint on each set of 3 consecutives locations which are

visited by a vehicle.

SOLUTION Allows to impose constraints on set of vertices of the

final graph:

• provide generators for set of vertices,
• restrictions: − polynomial number of sets (pred, succ, ...),

− computing each set can be done in polynomial time.

: TASKS: collection(origin-dvar,duration-dvar end-dvar, high-dvar) LIMIT: int : at least 2 out of [TASKS.origin,TASKS.duration,TASKS.end] required(TASKS.high) TASKS.duration ≥ 0 TASKS.high ≥ 0 LIMIT ≥ 0 : TASKS : IDENTITY : TASKS : CLIQUE : 2 : TASKS.origin[1]+TASKS.duration[1]=TASKS.end[1] ∧ TASKS.origin[2]+TASKS.duration[2]=TASKS.end[2] ∧ TASKS.origin[1]<TASKS.end[2] ∧ TASKS.origin[2]<TASKS.end[1] : NVERTEX = |TASKS| : CLIQUE(MAX) : Sum(TASKS.high) ≤ LIMIT

• ARGUMENT
• RESTRICTION(S)
• VERTEX INPUT
• VERTEX GENERATOR
• EDGE INPUT
• EDGE GENERATOR
• EDGE ARITY
• EDGE CONSTRAINT
• GRAPH PROPERTIE
• SETS
• DYNAMIC CONSTRAINT

Avoid Clique(MAX) as a set generator

Generates all maximum cliques on the final graph: in general can’t be computed in polynomial time !!!

Example of Use of Clique(MAX)

cumulative({

1 2 3 4 5 6 7 8 9 101112

≤ ≤ ≤ ≤ 8 1 2 3 4 5

• rigin-1 duration-3 end-4 high-1,
• rigin-2 duration-9 end-11 high-2,
• rigin-3 duration-10 end-13 high-1,
• rigin-6 duration-6 end-12 high-1,
• rigin-7 duration-2 end-9 high-3

}, 8)

Constraints Mentioning Clique(MAX)

• assign_and_count
• assign_and_nvalue
• interval_and_sum
• interval_and_count
• track
• bin_packing
• cumulative
• cumulatives
• cumulative_product
• coloured_cumulative
• cyclic_cumulative
• coloured_cumulatives
• cumulative_2d

OBSERVATION: all final graphs

associated to these constraints are interval graphs, so we use

• ne other model which avoids

Clique(MAX).

Intuition of the New Model

• ARC INPUT
• ARC GENERATOR
• ARC ARITY
• ARC CONSTRAINT
• SETS

: Time points, Tasks : product : 2 : overlap : succ Time points Tasks Is overlapped by

p1 p2 p3 p4 p5 p6 t1 t2 t3 {t1, t2} {t2, t3} {t3}

Which Time Points ?

The task origins and ends:

1 2 3 4 5 6 7 8 9 10 11 12

• But the time points are not explicitly mentioned in the constraint?

Generates a collection containing one single item

Derived Collections

An optional extra field for describing a global constraint. Purpose : generates collections from the parameters of the constraint. Motivation: allows to simplify parameters of a constraint.

FROM element(INDEX,TABLE,VALUE) INDEX : dvar TABLE: collection(index-dvar,value-dvar) VALUE: dvar TO element(ITEM,TABLE) ITEM : collection(index-dvar,value-dvar) TABLE: collection(index-int ,value-int ) DERIVED COLLECTION: SIMPLE ITEM: (index-INDEX, value-VALUE)

FIELD NAME COLLECTION GENERATOR NAME OF GENERATED COLLECTION PARAMETERS FOR THE COLLECTION GENERATOR

Generates a collection by applying an item generator to each item of the TASKS collection

Derived Collection for cumulatives

CONSTRAINT: cumulatives(TASKS,MACHINES,CTR) ARGUMENTS: TASKS : collection(machine-dvar,origin-dvar,duration-dvar,end-dvar,height-dvar) MACHINES: collection(id-int,capacity-int) CTR : atom ..........................................................................................

DERIVED COLLECTION: EXPAND(TASKS) TIME_POINTS: (duration-1, origin-origin, end-origin+1) (duration-1, origin-end , end-end+1 )

FIELD NAME COLLECTION GENERATOR NAME OF GENERATED COLLECTION PARAMETERS FOR THE COLLECTION GENERATOR

• DATA CONSTRAINTS
• VALUE CONSTRAINTS (one set of variables)
• VALUE CONSTRAINTS (two sets of variables)
• ORDERING CONSTRAINTS
• CHANNELING CONSTRAINTS
• GRAPH CONSTRAINTS
• RELAXATION AND PROXIMITY CONSTRAINTS

Categories of Global Constraints

ARGUMENTS :

ctr_arguments(element, [INDEX-dvar, TABLE-collection(value-dvar), VALUE-dvar]).

PURPOSE :

VALUE is equal to the INDEX-th item of TABLE.

Element

ORIGIN : [VanHentenryckCarillon88] EXAMPLE :

element(3, [[value-6],[value-9],[value-2],[value-9]], 2)

Element

ctr_graph(element, [ITEM,TABLE], 2, [PRODUCT>>collection(item,table)], [item.index = table.key, item.value = table.value], [NARC = 1]).

1:index value 2:key value

item.index = table.key item.value = table.value ARC CONSTRAINT PRODUCT ITEM TABLE INITIAL GRAPH

Element

PRODUCT ITEM TABLE INITIAL GRAPH

NARC=1

FINAL GRAPH element(3, [[value-6],[value-9],[value-2],[value-9]], 2)

ARGUMENTS :

ctr_arguments(symmetric_alldifferent, [NODES-collection(index-int, succ-dvar)]).

PURPOSE :

All variables associated to the succ attribute of the NODES collection should be pairwise distinct. In addition enforce the following condition: If variable NODES[i].succ takes value j then variable NODES[j].succ takes value i.

ORIGIN : [Régin99] EXAMPLE :

symmetric_alldifferent([[index-1, succ-3], [index-2, succ-4], [index-3, succ-1], [index-4, succ-2]])

Symmetric_alldifferent

Symmetric_alldifferent

NODES INITIAL GRAPH CLIQUE(≠ ≠ ≠ ≠) ctr_graph(symmetric_alldifferent, [NODES], 2, [CLIQUE(≠ ≠ ≠ ≠)>>collection(nodes1,nodes2)], [nodes1.succ = nodes2.index, nodes2.succ = nodes1.index], [NARC = size(NODES)]).

1:index succ 2:index succ

nodes1.succ = nodes2.index, nodes2.succ = nodes1.index ARC CONSTRAINT

Symmetric_alldifferent

NODES INITIAL GRAPH CLIQUE(≠ ≠ ≠ ≠) FINAL GRAPH

NARC=4

symmetric_alldifferent([[index-1, succ-3], [index-2, succ-4], [index-3, succ-1], [index-4, succ-2]])

ARGUMENTS :

ctr_arguments(balance, [BALANCE-dvar , VARIABLES-collection(var-dvar)]).

PURPOSE :

BALANCE is equal to the difference between the number of

• ccurrence of the value that occurs the most and the value that
• ccurs the least within the collection of variables VARIABLES.

Balance

ORIGIN : [Beldiceanu00] EXAMPLE :

balance(2,[[var-3],[var-1],[var-7],[var-1],[var-1]])

Balance: description

CLIQUE VARIABLES

=

INITIAL GRAPH ctr_graph(balance, [VARIABLES], 2, [CLIQUE>>collection(variables1,variables2)], [variables1.var = variables2.var], [RANGE_NSCC = BALANCE]).

1:var 2:var

variables1.var = variables2.var ARC CONSTRAINT

=

Balance: from initial to final graph

FINAL GRAPH

RANGE_NSCC=3-1=2

CLIQUE VARIABLES

=

INITIAL GRAPH balance(2,[[var-3],[var-1],[var-7],[var-1],[var-1]])

ARGUMENTS :

ctr_arguments(disjoint,[VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar)]).

PURPOSE :

Each variable of the collection VARIABLES1 should be different from all the variables of the collection VARIABLES2.

ORIGIN : N.Beldiceanu EXAMPLE :

disjoint([[var-1],[var-9],[var-1],[var-5]], [[var-2],[var-7],[var-7],[var-0],[var-6],[var-8]])

Disjoint

Disjoint

1:var 2:var

variables1.var = variables2.var ARC CONSTRAINT PRODUCT VARIABLES1 INITIAL GRAPH VARIABLES2 ctr_graph(disjoint, [VARIABLES1, VARIABLES2], 2, [PRODUCT>>collection(variables1, variables2)], [variables1.var = variables2.var], [NARC = 0]). FINAL GRAPH is empty (NARC = 0)

ARGUMENTS :

ctr_arguments(common,[NCOMMON1-dvar, NCOMMON2-dvar, VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar)]).

PURPOSE :

NCOMMON1 is the number of variables of the collection of variables VARIABLES1 taking a value in VARIABLES2. NCOMMON2 is the number of variables of the collection of variables VARIABLES2 taking a value in VARIABLES1.

ORIGIN : N.Beldiceanu EXAMPLE :

common(3,4,[[var-1],[var-9],[var-1],[var-5]], [[var-2],[var-1],[var-9],[var-9],[var-6],[var-9]])

Common

Common

INITIAL GRAPH FINAL GRAPH

ARGUMENTS :

ctr_arguments(used_by,[VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar)]).

PURPOSE :

All the values of the variables of collection VARIABLES2 are used by the variables of collection VARIABLES1.

ORIGIN : N.Beldiceanu EXAMPLE :

used_by([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[var-1],[var-1],[var-2],[var-5]])

Used_by

Used_by

For all CC: NSOURCE >= NSINK NSINK = size(VARIABLES2)

INITIAL GRAPH FINAL GRAPH

ARGUMENTS :

ctr_arguments(same, [VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar)]).

PURPOSE :

The variables of the collection VARIABLES2 correspond to the variables of the collection VARIABLES1 according to a permutation.

ORIGIN : N.Beldiceanu EXAMPLE :

same([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[var-9],[var-1],[var-1],[var-1],[var-2],[var-5]])

Same

Same

INITIAL GRAPH FINAL GRAPH

ARGUMENTS :

ctr_arguments(sort,[VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar)]).

PURPOSE :

The variables of the collection VARIABLES2 correspond to the variables of VARIABLES1 according to a permutation. The variables of VARIABLES2 are sorted in increasing order.

ORIGIN : [OlderSwinkelsEmden95] EXAMPLE :

sort([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[var-1],[var-1],[var-1],[var-2],[var-5],[var-9]])

Sort

Sort

PRODUCT VARIABLES1 INITIAL GRAPH VARIABLES2

1:var 2:var

variables1.var = variables2.var ARC CONSTRAINT ctr_graph(sort, [VARIABLES1, VARIABLES2], 2, [PRODUCT>>collection(variables1, variables2)], [variables1.var = variables2.var], [for_all(CC,NSOURCE = NSINK), NSOURCE = size(VARIABLES1), NSINK = size(VARIABLES2)]).

Sort

1:var 2:var

variables1.var ≤ variables2.var ARC CONSTRAINT ctr_graph(sort, [VARIABLES2], 2, [PATH>>collection(variables1, variables2)], [variables1.var ≤ variables2.var], [NARC = size(VARIABLES2)-1]). INITIAL GRAPH VARIABLES2 PATH

PRODUCT

VARIABLES1 INITIAL GRAPH VARIABLES2 FINAL GRAPH

CC#1: NSOURCE=3, NSINK=3 CC#2: NSOURCE=1, NSINK=1 CC#3: NSOURCE=1, NSINK=1 CC#4: NSOURCE=6, NSINK=6 NSOURCE=3, NSINK=3

Sort

sort([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[var-1],[var-1],[var-1],[var-2],[var-5],[var-9]])

ARGUMENTS :

ctr_arguments(sort_permutation, [FROM-collection(var-dvar), PERMUTATION-collection(ind-dvar), TO-collection(var-dvar)]).

PURPOSE :

The variables of collection FROM correspond to the variables

• f collection TO according to the permutation PERMUTATION.

The variables of collection TO are sorted in increasing order.

ORIGIN : [Zhou97] EXAMPLE :

sort_permutation([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[ind-1],[ind-6],[ind-3],[ind-5],[ind-4],[ind-2]], [[var-1],[var-1],[var-1],[var-2],[var-5],[var-9]])

Sort_permutation

Sort_permutation

PRODUCT FROM_PERMUTATION

INITIAL GRAPH

TO

ctr_graph(sort_permutation, [FROM_PERMUTATION,TO], 2, [PRODUCT>>collection( from_permutation,to)], [from_permutation.var = to.var, from_permutation.ind = to.key], [NARC = size(PERMUTATION)]).

1:var ind 2:key ind

from_permutation.var = to.var, from_permutation.ind = to.key ARC CONSTRAINT + a graph for ordering the items

• f the collection TO (like sort)

FINAL GRAPH

Sort_permutation

NARC=6 PRODUCT FROM_PERMUTATION

INITIAL GRAPH

TO

sort_permutation([[var-1],[var-9],[var-1],[var-5],[var-2],[var-1]], [[ind-1],[ind-6],[ind-3],[ind-5],[ind-4],[ind-2]], [[var-1],[var-1],[var-1],[var-2],[var-5],[var-9]])

ARGUMENTS :

ctr_arguments(symmetric_gcc, [VARS-collection(idvar-int, var-svar, nocc-dvar), VALS-collection(idval-int, val-svar, nocc-dvar)]).

PURPOSE :

Put in relation two sets: for each element of one set gives the corresponding elements of the other set to which it is associated. In addition enforce constraints

• n the number of occurrences of each value.

ORIGIN : Derived from global_cardinality by W.Kocjan. EXAMPLE :

symmetric_gcc([[idvar-1, var-{3} , nocc-1],[idvar-2, var-{1} , nocc-1], [idvar-3, var-{1,2} , nocc-2],[idvar-4, var-{1,3}, nocc-2]], [[idval-1, val-{2,3,4}, nocc-3],[idval-2, val-{3} , nocc-1], [idval-3, val-{1,4} , nocc-2],[idval-4, val-{} , nocc-0]])

Symmetric_gcc

Symmetric_gcc

PRODUCT VARS INITIAL GRAPH VALS ctr_graph(symmetric_gcc, [VARS,VALS], 2, [PRODUCT>>collection(vars,vals)], [in_set(vars.idvar,vals.val) ⇔ ⇔ ⇔ ⇔ in_set(vals.idval,vars.var), vars.nocc = card_set(vars.var), vals.nocc = card_set(vals.val)], [NARC = size(VARS)*size(VALS)]).

1:idvar var nocc 2:idval val nocc

ARC CONSTRAINT

PRODUCT VARS INITIAL GRAPH VALS FINAL GRAPH

NARC=16

symmetric_gcc([[idvar-1, var-{3} , nocc-1],[idvar-2, var-{1} , nocc-1], [idvar-3, var-{1,2} , nocc-2],[idvar-4, var-{1,3}, nocc-2]], [[idval-1, val-{2,3,4}, nocc-3],[idval-2, val-{3} , nocc-1], [idval-3, val-{1,4} , nocc-2],[idval-4, val-{} , nocc-0]])

Symmetric_gcc

3 12 3 3 2 1

ARGUMENTS :

ctr_arguments(minimum,[MIN-dvar, VARIABLES-collection(var-dvar)]).

PURPOSE :

MIN is the minimum value of the collection of domain variables VARIABLES.

ORIGIN : CHIP EXAMPLE :

minimum(2,[[var-3],[var-2],[var-7],[var-2],[var-6]])

Minimum

Minimum

CLIQUE VARIABLES INITIAL GRAPH ctr_graph(minimum, [VARIABLES], 2, [CLIQUE>>collection(variables1, variables2)], [variables1 = variables2 \/ variables1.var < variables2.var], [ORDER(0,MAXINT,var) = MIN]).

1:var 2:var

variables1 = variables2 \/ variables1.var < variables2.var ARC CONSTRAINT

Minimum

FINAL GRAPH

ORDER(0,MAXINT,var)=2

CLIQUE VARIABLES INITIAL GRAPH minimum(2,[[var-3],[var-2],[var-7],[var-2],[var-6]]) 2 2

ARGUMENTS :

ctr_arguments(lex_lesseq, [VECTOR1-collection(var-dvar), VECTOR2-collection(var-dvar)]).

PURPOSE :

VECTOR1 is lexicographically less than or equal to VECTOR2.

Lex_lesseq

ORIGIN : CHIP EXAMPLE :

lex_lesseq([[var-5], [var-2], [var-3], [var-1]], [[var-5], [var-2], [var-6], [var-2]])

Lex_lesseq

PRODUCT(PATH,VOID)

COMPONENTS INITIAL GRAPH DESTINATION

= = < < <

≤ ≤ ≤ ≤

ctr_graph(lex_lesseq, [COMPONENTS,DESTINATION], 2, [PRODUCT(PATH,VOID)>> collection(item1,item2)], [(item2.index>0 /\ item1.x = item1.y) \/ (item1.index < size('VECTOR1') /\ item2.index=0 /\ item1.x < item1.y) \/ (item1.index = size('VECTOR1') /\ item2.index=0 /\ item1.x ≤ item1.y], [PATH_FROM_TO(index,1,0) = 1]).

1:index x y 2:index x y

ARC CONSTRAINT item1.x = item1.y item1.x < item1.y item1.x ≤ item1.y

=

Lex_lesseq

PRODUCT(PATH,VOID)

COMPONENTS INITIAL GRAPH DESTINATION

= = < < <

≤ ≤ ≤ ≤

FINAL GRAPH

PATH_FROM_TO(index,1,0)

=

1:0,0,0 1:1,5,5

COMPONENTS DESTINATION

5=5 2=2 3<6 1≤ ≤ ≤ ≤2

ARGUMENTS :

ctr_arguments(domain_constraint, [VAR-dvar, VALUES-collection(var01-dvar, value-int)]).

PURPOSE :

Used in order to make the link between a formulation using finite domain constraints and a formulation based on linear programming.

Domain_constraint

ORIGIN : [Refalo00] EXAMPLE :

domain_constraint(5,[[var01-0, value-9], [var01-1, value-5], [var01-0, value-2], [var01-0, value-7]])

Domain_constraint

ctr_graph(domain_constraint, [VALUE,VALUES], 2, [PRODUCT>>collection(value,values)], [(value.value=values.value /\ values.var01=1) \/ (value.value≠ ≠ ≠ ≠values.value /\ values.var01=0)], [NARC = size(VALUES)]).

1:var01 value 2:var01 value

ARC CONSTRAINT

(value.value=values.value /\ values.var01=1) \/ (value.value≠ ≠ ≠ ≠values.value /\ values.var01=0)

PRODUCT VALUE VALUES INITIAL GRAPH

Domain_constraint

PRODUCT VALUE VALUES INITIAL GRAPH FINAL GRAPH

NARC=4

domain_constraint(5,[[var01-0, value-9], [var01-1, value-5], [var01-0, value-2], [var01-0, value-7]])

ARGUMENTS :

ctr_arguments(link_set_to_booleans, [SVAR-svar, BOOLEANS-collection(bool-dvar, val-int)]).

PURPOSE :

Make the link between a set variable SVAR and those 0-1 variables that are associated to each potential value belonging to SVAR.

ORIGIN : Inspired by domain_constraint. EXAMPLE :

link_set_to_booleans({1,3,4}, [[bool-0, val-0],[bool-1, val-1],[bool-0, val-2], [bool-1, val-3],[bool-1, val-4],[bool-0, val-5]])

Link_set_to_booleans

Link_set_to_booleans

FINAL GRAPH INITIAL GRAPH

ARGUMENTS :

ctr_arguments(cutset, [SIZE_CUTSET-dvar, NODES-collection(index-int, succ-sint, bool-dvar)]).

PURPOSE :

Consider a directed graph G with n vertices. Enforces that the subset of kept vertices of cardinality n-SIZE_CUTSET and their corresponding arcs form a graph without circuit.

Cutset

ORIGIN : [FagesLal03] EXAMPLE :

cutset(1, [[index-1, succ-{2,3,4}, bool-1], [index-2, succ-{3}, bool-1], [index-3, succ-{4}, bool-1], [index-4, succ-{1}, bool-0]])

Cutset: description

NODES

INITIAL GRAPH ctr_graph(cutset, [NODES], 2, [CLIQUE>>collection(nodes1,nodes2)], [nodes2.index∈

∈ ∈ ∈nodes1.succ,

nodes1.bool = 1 , nodes2.bool = 1 ], [MAX_NSCC ≤ 1, NVERTEX = size(NODES) − SIZE_CUTSET]).

1:index

succ bool

nodes2.index∈

∈ ∈ ∈nodes1.succ,

nodes1.bool = 1, nodes2.bool = 1 ARC CONSTRAINT

2:index

succ bool CLIQUE

cutset(1, [[index-1, succ-{2,3,4}, bool-1], [index-2, succ-{3}, bool-1], [index-3, succ-{4}, bool-1], [index-4, succ-{1}, bool-0]])

FINAL GRAPH

Cutset: from initial to final graph

MAX_NSCC=1, NVERTEX=3

INITIAL GRAPH

NODES CLIQUE

ARGUMENTS :

ctr_arguments(clique, [SIZE_CLIQUE-dvar, NODES-collection(index-int, succ-svar)]).

PURPOSE :

Consider a directed graph G. Select a subset of the vertices of G which form a clique of size SIZE_CLIQUE.

Clique

ORIGIN : [Fahle02] EXAMPLE :

clique(3, [[index-1, succ-{}], [index-2, succ-{3,5}], [index-3, succ-{2,5}], [index-4, succ-{}], [index-5, succ-{2,3}]])

Clique

NODES

INITIAL GRAPH ctr_graph(clique, [NODES], 2, [CLIQUE >>collection(nodes1,nodes2)], [nodes1 ≠ nodes2, in_set(nodes2.index,nodes1.succ)], [NARC = SIZE_CLIQUE*SIZE_CLIQUE −SIZE_CLIQUE, NVERTEX = SIZE_CLIQUE]).

1:index

succ

nodes1 ≠ nodes2, in_set(nodes2.index,nodes1.succ) ARC CONSTRAINT

2:index

succ CLIQUE

Clique

FINAL GRAPH

NARC=6, NVERTEX=3 NODES

INITIAL GRAPH clique(3, [[index-1, succ-{}], [index-2, succ-{3,5}], [index-3, succ-{2,5}], [index-4, succ-{}], [index-5, succ-{2,3}]])

CLIQUE

ARGUMENTS :

ctr_arguments(k_cut, [K-int, NODES-collection(index-int, succ-svar)]).

PURPOSE :

Select some arcs of a directed graph in order to have at least K connected components (an isolated vertex is counted as one connected component).

K_cut

ORIGIN : E. Althaus EXAMPLE :

k_cut(3,[[index-1, succ-{}],[index-2, succ-{3,5}],[index-3, succ-{5}], [index-4, succ-{}],[index-5, succ-{2,3}]])

K_cut

ctr_graph(k_cut, [NODES], 2, [CLIQUE >>collection(nodes1,nodes2)], [nodes1=nodes2 \/ in_set(nodes2.index,nodes1.succ)], [NCC >= K]).

1:index

succ

nodes1=nodes2 \/ in_set(nodes2.index,nodes1.succ) ARC CONSTRAINT

2:index

succ CLIQUE NODES

INITIAL GRAPH

K_cut

CLIQUE NODES

INITIAL GRAPH FINAL GRAPH

NCC=3

k_cut(3,[[index-1, succ-{}],[index-2, succ-{3,5}],[index-3, succ-{5}], [index-4, succ-{}],[index-5, succ-{2,3}]])

ARGUMENTS :

ctr_arguments(path_from_to,[FROM-int, TO-int, NODES-collection(index-int, succ-svar)]).

PURPOSE :

MIN is the minimum value of the collection of domain variables VARIABLES.

ORIGIN : [AlthausBockmayrElfKasperJungerMehlhorn02] EXAMPLE :

path_from_to(4,3,[[index-1, succ-{} ],[index-2, succ-{}],[index-3, succ-{5}], [index-4, succ-{5}],[index-5, succ-{2,3}]])

Path_from_to

Path_from_to

CLIQUE NODES

INITIAL GRAPH ctr_graph(path_from_to, [NODES], 2, [CLIQUE >>collection(nodes1,nodes2)], [in_set(nodes2.index,nodes1.succ)], [PATH_FROM_TO(index,FROM,TO) = 1]).

1:index

succ

in_set(nodes2.index,nodes1.succ) ARC CONSTRAINT

2:index

succ

Path_from_to

FINAL GRAPH

PATH_FROM_TO(index,4,3)=1 CLIQUE NODES

INITIAL GRAPH path_from_to(4,3,[[index-1, succ-{} ],[index-2, succ-{}],[index-3, succ-{5}], [index-4, succ-{5}],[index-5, succ-{2,3}]])

! !"#

ARGUMENTS :

ctr_arguments(soft_alldifferent_ctr, [C-dvar, VARIABLES-collection(var-dvar)]).

PURPOSE :

Consider the disequality constraints involving two distinct variables of the collection

• VARIABLES. Among the previous set of constraints,

C is the number of disequality constraints which do not hold.

ORIGIN : [PetitRéginBessière01] EXAMPLE :

soft_alldifferent_ctr(4,[[var-5],[var-1],[var-9],[var-1],[var-5],[var-5]])

Soft_alldifferent_ctr

Soft_alldifferent_ctr

ctr_graph(soft_alddifferent_ctr, [VARIABLES], 2, [CLIQUE(<)>>collection(variables1, variables2)], [variables1.var = variables2.var], [NARC = C]).

1:var

variables1.var = variables2.var ARC CONSTRAINT

2:var

CLIQUE(<) VARIABLES

INITIAL GRAPH

Soft_alldifferent_ctr

CLIQUE(<) VARIABLES

INITIAL GRAPH FINAL GRAPH

NARC=4

soft_alldifferent_ctr(4,[[var-5],[var-1],[var-9],[var-1],[var-5],[var-5]])

ARGUMENTS :

ctr_arguments(soft_alldifferent_var,[C-dvar, VARIABLES-collection(var-dvar)]).

PURPOSE :

C is the minimum number of variables of the collection VARIABLES for which the value needs to be changed in order that all variables

• f VARIABLES take a distinct value.

ORIGIN : [PetitRéginBessière01] EXAMPLE :

soft_alldifferent_var(3,[[var-5],[var-1],[var-9],[var-1],[var-5],[var-5]])

Soft_alldifferent_var

Soft_alldifferent_var

CLIQUE VARIABLES

INITIAL GRAPH ctr_graph(soft_alddifferent_var, [VARIABLES], 2, [CLIQUE >>collection(variables1, variables2)], [variables1.var = variables2.var], [size(VARIABLES) - NSCC = C]).

1:var

variables1.var = variables2.var ARC CONSTRAINT

2:var

Soft_alldifferent_var

CLIQUE VARIABLES

INITIAL GRAPH FINAL GRAPH

NSCC=3

soft_alldifferent_var(3,[[var-5],[var-1],[var-9],[var-1],[var-5],[var-5]])

ARGUMENTS :

ctr_arguments(distance_between, [DIST-dvar, VARIABLES1-collection(var-dvar), VARIABLES2-collection(var-dvar), CTR-[=,=\=,<,>=,>,=<]).

PURPOSE :

Let Ui and Vi be respectively the i-th and j-th variables of VARIABLES1. Let Xi and Yi be respectively the i-th and j-th variables of VARIABLES2. DIST is equal to the number of times one of the following mutually incompatible conditions are true: Ui CTR Vi holds and Xi CTR Yi does not hold, Xi CTR Yi holds and Ui CTR Vi does not hold.

ORIGIN : N.Beldiceanu. EXAMPLE :

distance_between(2,[[var-3],[var-4],[var-6],[var-2],[var-4]], [[var-2],[var-6],[var-9],[var-3],[var-6]],<)

Distance_between

Distance_between

Exercise: modelling the non-overlapping constraint between rectangles

Model the non-overlapping constraint between n rectangles. Each rectangle is described by its origin (lower leftmost point), its sizes and its ends (all domain variables):

• ox: coordinate of its origin on x,
• oy: coordinate of its origin on y,
• sx: size on x,
• sy: size on y,
• ex: end (ox+sx) on x,
• ey: end (oy+sy) on y.

Hints: First model the non-overlapping constraint between two rectangles, (use the arc generator SYMMETRIC_PRODUCT(=), take care of the case when size can be equal to 0) Then model the non-overlapping constraint between n rectangles (use CLIQUE(≠))