sweconsnet, may 24 2002

SweConsNet, May 24 2002

Activities of the Combinatorial Problem Solving Group

Nicolas Beldiceanu

SICSLägerhyddsvägen 18

75237 Uppsala

email: [email protected]

OVERVIEW

CLASSIFICATION OF GLOBAL CONSTRAINTS

GENERIC FILTERING ALGORITHMS

SWEEP BASED FILTERING ALGORITHMS

RESOURCES CONSTRAINT

COST FILTERING ALGORITHMS

ENGINEERING OF FILTERING ALGORITHMS

CONSTRAINT DEBUGGING

IMPLIED CONSTRAINTS

SUMMARY AND CONCLUSION

Outline

OVERVIEW








IMPLIED CONSTRAINTS


Short Term Focus

Explicit description of constraints

Efficient and generic filtering algorithms

Long Term Perspective

Synthetize filtering algorithms from the description of constraints

How to Proceed

Interdisciplinary interactions (Applied Math, Algorithmic, Constraint)

Implementation (make it available in SICStus Prolog)

Feedback from applications (configuration, scheduling, biology,

Overview

electronic market)

Overview

People

N.Beldiceanu, M.Carlsson, P.Mildner, T.Szeredi, M.Ågren

Collaborations

SICS Kista (M.Aronsson, E.Aurell, P.Kreuger)

MPII Saarbrücken (S.Thiel)

Uppsala University (Q.Guo)

OVERVIEW








IMPLIED CONSTRAINTS


Motivations for a Classification of Global Constraints

• Find out the basic constituents of the global constraint,

• Classify the properties of each basic constituent,

• Understand how properties interact.

Main Idea of the Classification

Global Constraints as:

Graph Properties

on Structured Network

of Elementary Constraints

of the Same Type

nvalue(NVAL, VARIABLES )

• A R G U M E N T

• R E S T R I C T I O N (S)

• V E R T E X I N P U T

• V E R T E X G E N E R A T O R

• E D G E I N P U T

• E D G E G E N E R A T O R

• E D G E A R I T Y

• E D G E C O N S T R A I N T

• G R A P H P R O P E R T Y

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

: NVAL : dvar

VARIABLES: collection(var-dvar)

: NVAL 0

NVAL |VARIABLES|

required(VARIABLES.var)

: VARIABLES

: IDENTITY

: VARIABLES

: CLIQUE

: 2

: VARIABLES.var[1] = VARIABLES.var[2]

: NSCC = NVAL

Collections of items:

VARIABLES

Vertices generator:

IDENTITY

Edge generator:

CLIQUE

V1

V2

V3

V4V5

V1 V2 V3 V4 V5

V1

V2

V3

V4V5Edge constraint:

=

V1

V2

V3

V4V5

= =

=

=

=

== =

= =

=

= =

==

3

1

7

16

==

= =

==

Graph property:

NSCC=NVAL

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

A Catalog of Global Constraints

Alldifferent

Alldifferent_except_0

Alldifferent_interval

Alldifferent_modulo

Alldifferent_partition

Alldifferent_same_value

Among

Among_interval

Among_modulo

Among_seq

Assign_and_count

Assign_and_nvalue

Balance

Balance_modulo

Balance_partition

Bin_packing

Binary_tree

Cardinality_atleast

Cardinality_atmost

Change

Change_continuity

Change_pair

Change_partition

Circuit

Circuit_cluster

Circular_change

Coloured_cumulative

Coloured_cumulatives

Common

Common_interval

Common_modulo

Common_partition

Connect_points

Connected

Count

Crossing

Cumulative

Cumulative_2d

Cumulative_product

Cumulatives

Cycle

Cycle_card_on_path

Cycle_cover

Cycle_or_accessibility

Cycle_resource

Cyclic_change

Cyclic_change_joker

Cyclic_cumulative

Derangement

Diffn

Diff_2

Diff_2_cyclic

Diff_2_min_dist

Disjoint

Disjoint_tasks

Distance_change

Distance_less

Distribute

Domain_constraint

Element

Element_greatereq

Element_lesseq

Element_sparse

Elements

Elements_alldifferent

Global_cardinality

Golomb

Graph_crossing

Group

Group_skip_isolated_item

Inflexion

Interval_and_count

Interval_and_sum

Inverse

Longest_change

Map

Max_index

Max_n

Max_nvalue

Maximum

Maximum_modulo

Maximum_pair

Min_index

Min_n

Min_nvalue

Minimum

Minimum_except_0

Minimum_modulo

Minimum_pair

Nclass

Nequivalence

Ninterval

Notallequal

Npair

Number_of_rest

Nvalue

Orchad

Place_in_pyramid

Polyomino

Relaxed_sliding_sum

Same

Same_interval

Same_modulo

Same_partition

Sliding_card_skip0

Sliding_sum

Sliding_time_window

Smooth

Soft_alldifferent_ctr

Soft_alldifferent_var

A Catalog of Global Constraints

Stretch

Stretch_circuit

Stretch_path

Symmetric_alldiff

Temporal_path

Tree

Tree_resource

Used_by

Used_by_interval

Used_by_modulo

Used_by_partition

Old constraints which were not inside: element, ...

Invented constraints: disjoint, ...

Catalog updated as new constraints are presented

domain_constraint [REFALO, CP2000]

stretch [PESANT, CP2001]

soft alldifferent [PETIT, RÉGIN, CP2001]

Modifications in the way of describing constraints

possibility to have constraint (like global cardinality, stretch) where

different limits are associated to different values

avoid properties which can’t be evaluated in polynomial time when all

parameters of the constraint are fixed (e.g maximum clique)

OVERVIEW








IMPLIED CONSTRAINTS


Filtering Algorithms for Families of Constraints

Alldifferent

Alldifferent_except_0

Alldifferent_interval

Alldifferent_modulo

Alldifferent_partition

Alldifferent_same_value

Among

Among_interval

Among_modulo

Among_seq

Assign_and_count

Assign_and_nvalue

Balance

Balance_modulo

Balance_partition

Bin_packing

Binary_tree

Cardinality_atleast

Cardinality_atmost

Change

Change_continuity

Change_pair

Change_partition

Circuit

Circuit_cluster

Circular_change

Coloured_cumulative

Coloured_cumulatives

Common

Common_interval

Common_modulo

Common_partition

Connect_points

Connected

Count

Crossing

Cumulative

Cumulative_2d

Cumulative_product

Cumulatives

Cycle

Cycle_card_on_path

Cycle_cover

Cycle_or_accessibility

Cycle_resource

Cyclic_change

Cyclic_change_joker

Cyclic_cumulative

Derangement

Diffn

Diff_2

Diff_2_cyclic

Diff_2_min_dist

Disjoint

Disjoint_tasks

Distance_change

Distance_less

Distribute

Domain_constraint

Element

Element_greatereq

Element_lesseq

Element_sparse

Elements

Elements_alldifferent

Global_cardinality

Golomb

Graph_crossing

Group

Group_skip_isolated_item

Inflexion

Interval_and_count

Interval_and_sum

Inverse

Longest_change

Map

Max_index

Max_n

Max_nvalue

Maximum

Maximum_modulo

Maximum_pair

Min_index

Min_n

Min_nvalue

Minimum

Minimum_except_0

Minimum_modulo

Minimum_pair

Nclass

Nequivalence

Ninterval

Notallequal

Npair

Number_of_rest

Nvalue

Orchad

Place_in_pyramid

Polyomino

Relaxed_sliding_sum

Same

Same_interval

Same_modulo

Same_partition

Sliding_card_skip0

Sliding_sum

Sliding_time_window

Smooth

Soft_alldifferent_ctr

Soft_alldifferent_var

OVERVIEW








IMPLIED CONSTRAINTS


Applications of Sweep Algorithms

Within the Geometry Literature Database, more than 100 references:

Voronoi diagram

Map overlay

Nearest objects

Triangulations

Hidden surface removals

Rectangles intersection

Shortest path

But not yet used within constraint programming !

y

x

event point

sweep line

sweep line status

(1)

(2)

(1) Move to next event point (2) Update sweep line status

Applications of Sweep Algorithms within Constraint Programming

Pruning for the following constraint patterns:

A conjunction of constraints with two shared variables

The cardinality operator with two shared variables

The non-overlapping constraint between polygons

A multi-resource cumulatives constraint

Several conjunction of constraints with one shared variable

Examples of Forbidden and Safe Regions

0X40Y40R9

X

Y

0 1 2 3 401234

X

Y

0 1 2 3 401234

X

Y

0 1 2 3 401234

0X40Y42Z3

0X4 0Y40T0 1U2

X

Y

0 1 2 3 401234

0X4 0Y4

alldifferent(

|X-Y|>Z

X

Y

0 1 2 3 401234

0X4 0Y41S6

X+2YS

X+1T

T+1X

Y+1U

U+4Y X+Y0 (mod 2){X,Y,4-Y,R})

alldifferent({X,Y,4-Y,R})An Example

0X4 0Y4 2Z3

1S6 0T0

1U2

alldifferent({X,Y,4-Y, R})

|X-Y|>Z

X+2YS

X+1T T+1X

Y+1U U+4Y

X+Y0 (mod 2)

PROBLEM:

Adjust minimum of X

according to Y and to the

fact that 4 or 5 constraints

should hold:

|X-Y|>Z

X+2YS

X+1T T+1X Y+1U U+4Y

X+Y0 (mod 2)

Y

X=001234

Y

01234

X=2

Y

01234

X=1

Deduction:

X>1

OVERVIEW








IMPLIED CONSTRAINTS


The cumulative Constraint

The original cumulative constraint[Aggoun & Beldiceanu 92]:

Restrict the resource consumptionat each point in time.

The generalized cumulatives constraint[Beldiceanu & Carlsson 01]:

A pool of cumulative resources ,

Height of a task can be negative ,

Maximum or minimum resource consumption ,

Holds for time-points crossed by at least one task.

1 2 3 4 5 6

01

2

34

12

3 4

time

resource consumption 4

time

resource consumption

1 2 3 4 5 6-2

-1

0

1

2

-1

0

1

Cumulatedprofiles

2

13

45

7

6 0

0

time

resource consumption

-2

-1

0

1

2

-1

0

1

3

5

7

6 0

0

Machine 1

Machine 2

1

2

4

For all tasks :

For all time-points crossed by task :

Let be the machine where task is assigned ,

For all tasks which both cross point , and are assigned to

:The sum of the height of tasks is not greater (less)

than the capacity of machine .

DEFINITION

1 2 3 4 5 6

t

t

t

i

m

s i m

m

EXAMPLE

s

OVERVIEW








IMPLIED CONSTRAINTS


Include Objective Within Constraint

12,

76

,32

,51

,1,6,1

uesstinct_valghts_of_disum_of_wei

weightval

weightval

weightval

varvarvar

Algorithms

sum_of...(Variables, Values, Cost)

FILTERING ALGORITHMS

find BOUNDSfor Cost

PROPAGATE from boundsof Cost to Variables

Lower bound LB:domination

Upper bound UB:matching

Remove val from var iff:LB+lower_regret(var,val)>max(Cost)

Remove val from var iff:UB-upper_regret(var,val)<min(Cost)

Algorithms (main results)

Lower bound LB:domination

Upper bound UB:matching

O(n log n + m)for a tight bound(when intervals)n:number of variablesm: number of values

O(n log n + m)for computing all the exact lower regret

of all values (when intervals)n:number of variablesm: number of values

O(m log m + c e)for a tight bound

m: number of values c:cardinality of max.matchinge:nb.of edges in bipartite graph

O(e)for computing all the exact upper regret

e:number of edges in bipartite graph

OVERVIEW








IMPLIED CONSTRAINTS


Engineering

Integrate standard constraints from other systems

Global cardinality, sort, ...

Reuse algorithms from literature

Alldifferent [Mehlhorn, Thiel], Knapsack (dynamic programming [Trick])

Incremental algorithms

Special cases when go down, optimizations valid for several global constraints (target, source)

Reingenering of the code

Revisiting filtering algorithms for inserting explanations

Making the system available on several platforms

OVERVIEW








IMPLIED CONSTRAINTS


Different Aspects of Trace

Control of Execution

Posting constraints, demons, waking constraints, entailment, failure, choice points

Locating Information

Situate a piece of information according to the context it originally came

Domain Modification

Intended pruning, domain modification, domain

Declarative Aspect

Explanation(s) for pruning, failing, adding a constraint, waking a demon

Procedural Aspect

Method used or not, nested block structure, method dependant explanations

OVERVIEW








IMPLIED CONSTRAINTS


Implied Constraints (as relations between graph properties)

11toofmaxadjust2

NCCNCCNVNA

2

14412toofmaxadjust

NVNANVNCC

11 2 NCCNCCNVNA

For a directed graph, relation between: NV : number of vertices, NA : number of arcs, NCC: number of connected components.

QUESTION

ANSWER

PROPAGATION RULESThe maximum number of arcs is achieved by having a complete clique in each connected components and the minimumnumber of connected components.

INTUITION

Implied Constraints (as relations between graph properties)

For a directed graph, relation between: NV : number of vertices, NA : number of arcs, NCC : number of connected components, NSCC: number of strongly connected components.

QUESTION

2

1111

NCCNSCCNCCNSCCNCCNVNSCCNVNCCNA

GUESS

• Provide a Constraint Classification

• Generic Propagation Algorithms for several Families of Constraints

• Use Algorithms from Graph and Geometry

• Several Constaints Implemented within SICStus

and Used within Demonstrators

Summary and Conclusion

Collaboration required with: applied mathematics, design of algorithms

Further Sources of Information

Papers (some at http://www.sics.se/libindex.html) Sweep as a Generic Pruning Technique Applied to the Non-Overlapping Rectangles

Constraint, [Beldiceanu,Carlsson 2001] .

Sweep as a Generic Pruning Technique Applied to Constraint Relaxation,

[Beldiceanu,Carlsson 2001] .

Non-overlapping Constraints between Convex Polytopes, [Beldiceanu,Guo,Thiel 2001] .

A New Multi-Resource cumulatives Constraint with Negative Heights,

[Beldiceanu,Carlsson 2001] .

Sweep Synchronization as a Global Propagation Mechanism [Beldiceanu,Carlsson,

Thiel 2002] . (submitted)

Cost-Filtering Algorithms for the two Sides of the Sum of Weights of Distinct Values

Constraint, [Beldiceanu,Thiel 2002] . (in preparation)

Implementation (http://www.sics.se/sicstus/) Alldifferent, Assignment, Case, Circuit, Count, Cumulative, Cumulatives, Disjoint1,

Disjoint2, Element, Global_cardinality, Knapsack, Scalar_product, Serialized, Sort,

Sum, Sum_of_weights_of_distinct_values.

sweconsnet, may 24 2002

Documents

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