Constraint Programming:In Pursuit of the Holy Grail

Roman Barták

Charles University, Praguebartak@kti.mff.cuni.cz

© Roman Barták, 1999

Talk ScheduleBasic notionsHistorical contextConstraint technology

– constraint satisfaction– constraints optimisation– over-constrained problems

ApplicationsSummary

– Advantages & Limitations

TrendsResources

© Roman Barták, 1999

What is CP?CP = Constraint Programming

– stating constraints about the problem variables– finding solution satisfying all the constraints

constraint = relation among several unknowns

Example: A+B=C, X>Y, N=length(S) …Features:

– express partial information X>2

– heterogeneous N=length(S)

– non-directional X=Y+2: X Y+2 YX-2

– declarative manner “

– additive X>2,X<5 X<5,X>2

– rarely independent A+B=5, A-B=1

© Roman Barták, 1999

The Origins

Artificial Intelligence– Scene Labelling (Waltz)

Interactive Graphics– Sketchpad (Sutherland)– ThingLab (Borning)

Logic Programming– unification --> constraint solving

Operations Research– NP-hard combinatorial problems

© Roman Barták, 1999

Scene Labellingfirst constraint satisfaction problemTask:

recognise objects in 3D scene by interpreting lines in 2D drawings

Waltz labelling algorithm– legal labels for junctions only– the edge has the same label at both ends

+

+

+ -

+

- ++ +

+

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Interactive GraphicsSketchpad (Sutherland)ThingLab (Borning)

– allow to draw and manipulate constrained geometric figures in the computer display

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Solving Technology

Constraint Satisfaction– finite domains -> combinatorial problems– 95% of all industrial applications

Constraints Solving– infinite or more complex domains– methods

• automatic differentiation, Taylor series, Newton method

– many mathematicians deal with whether certain constraints are satisfiable(Fermat’s Last Theorem)

© Roman Barták, 1999

Constraint Satisfaction ProblemConsist of:

– a set of variables X={x1,…,xn}

– variables’ domains Di (finite set of possible values)

– a set of constraints

Example:• X::{1,2}, Y::{1,2}, Z::{1,2}• X = Y, X Z, Y > Z

Solution of CSP– assignment of value from its domain to every variable

satisfying all the constraints

Example:• X=2, Y=2, Z=1

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Systematic Search Methods

exploring the solution spacecomplete and soundefficiency issues

Backtracking (BT)Generate & Test (GT)

exploringindividual assignments

exploring subspace

X

Z

Y

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GT & BT - ExampleProblem:

X::{1,2}, Y::{1,2}, Z::{1,2}

X = Y, X Z, Y > Z

generate & test backtrackingX Y Z test1 1 1 fail1 1 2 fail1 2 1 fail1 2 2 fail2 1 1 fail2 1 2 fail2 2 1 passed

X Y Z test1 1 1 fail

2 fail2 fail

2 1 fail2 1 passed

Systematic Search Methods

© Roman Barták, 1999

Consistency Techniquesremoving inconsistent values from variables’

domainsgraph representation of the CSP– binary and unary constraints only (no problem!)– nodes = variables– edges = constraints

node consistency (NC)arc consistency (AC)path consistency (PC)(strong) k-consistency

A

B

C

A>5

AB

A<C

B=C

© Roman Barták, 1999

Arc Consistency (AC)

the most widely used consistency technique (good simplification/performance ratio)

deals with individual binary constraints

repeated revisions of arcsAC-3, AC-4, Directional AC

Consistency Techniques

a

b

c

a

b

c

X Y

a

b

c

Z

© Roman Barták, 1999

AC - ExampleProblem:

X::{1,2}, Y::{1,2}, Z::{1,2} X = Y, X Z, Y > Z

1 2

1 2

1 2

1 2

1 2

1 2

Consistency Techniques

X

Y

Z

X

Y

Z

© Roman Barták, 1999

Is AC enough?empty domain => no solutioncardinality of all domains is 1 => solutionProblem:

X::{1,2}, Y::{1,2}, Z::{1,2} X Y, X Z, Y Z

In general, consistency techniques are incomplete!

1 2

1 2

1 2

Consistency Techniques

X

YZ

© Roman Barták, 1999

Constraint Propagationsystematic search only => no efficientconsistency only => no completeResult: combination of search (backtracking)

with consistency techniquesmethods:– look back (restoring from conflicts)– look ahead (preventing conflicts)

look back

Labelling order

look ahead

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Look Ahead - ExampleProblem:

X::{1,2}, Y::{1,2}, Z::{1,2}

X = Y, X Z, Y > Z

generate & test - 7 stepsbacktracking - 5 stepspropagation - 2 steps

X Y Z action result1 labelling

{1} {} propagation fail2 labelling

{2} {1} propagation solution

Constraint Propagation

© Roman Barták, 1999

Stochastic and Heuristic MethodsGT + smart generator of complete valuationslocal search - chooses best neighbouring configuration

– hill climbing neighbourhood = value of one variable changed

– min-conflicts neighbourhood = value of selected conflicting variable changed

avoid local minimum => noise heuristics– random-walk

sometimes picks next configuration randomly– tabu search

few last configurations are forbidden for next step

does not guarantee completeness

© Roman Barták, 1999

Connectionist approachArtificial Neural Networks

– processors (cells) = <variable,value>on state means “value is assigned to the variable”

– connections = inhibitory links between incompatible pairs

GENETstarts from random configuration

re-computes states using neighbouring cells

till stable configuration found (equilibrium)

learns violated constraints by strengthening weights

Incomplete (oscillation)

X Y

1

2

Z

variables

valu

es

© Roman Barták, 1999

Constraint Optimisationlooking for best solutionquality of solution measured by application

dependent objective function

Constraint Satisfaction Optimisation Problem– CSP– objective function: solution -> numerical value

Note: solution = complete labelling satisfying all the constraints

Branch & Bound (B&B)– the most widely used optimisation algorithm

© Roman Barták, 1999

Over-Constrained ProblemsWhat solution should be returned when

no solution exists?impossible satisfaction of all constraints

because of inconsistency Example: X=5, X=4

Solving methods– Partial CSP (PCSP)

weakening original CSP– Constraint Hierarchies

preferential constraints

© Roman Barták, 1999

Dressing Problem

shirt: {red, white}

footwear: {cordovans, sneakers}

trousers: {blue, denim, grey}

shirt x trousers: red-grey, white-blue, white-denim

footwear x trousers: sneakers-denim, cordovans-grey

shirt x footwear: white-cordovans

Over-Constrained Problems

red white

blue denim grey

cordovans sneakers

shirt

trousers footwear

© Roman Barták, 1999

Partial CSPweakening a problem:

– enlarging the domain of variable– enlarging the domain of constraint – removing a variable– removing a constraint

one solutionwhite - denim - sneakers

Over-Constrained Problems

enlarged constraint’s domain

red white

blue denim grey

cordovans sneakers

shirt

trousers footwear

© Roman Barták, 1999

Constraint Hierarchiesconstraints with preferencessolution respects the hierarchy

– weaker constraints do not cause dissatisfaction of stronger constraint

shirt x trousers @ requiredfootwear x trousers @ strongshirt x footwear @ weak

two solutionsred - grey - cordovans

white - denim - sneakers

Over-Constrained Problems

red white

blue denim grey

cordovans sneakers

shirt

trousers footwear

© Roman Barták, 1999

Applicationsassignment problems

– stand allocation for airports– berth allocation to ships– personnel assignment

• rosters for nurses

• crew assignment to flights

network management and configuration– planning of cabling of telecommunication networks– optimal placement of base stations in wireless networks

molecular biology– DNA sequencing

analogue and digital circuit design

© Roman Barták, 1999

Scheduling Problemsthe most successful application area

production scheduling (InSol Ltd.)well-activity scheduling (Saga Petroleum) forest treatment schedulingplanning production of jets (Dassault Aviation)

© Roman Barták, 1999

Advantages

declarative nature– focus on describing the problem to be solved, not on

specifying how to solve it

co-operative problem solving– unified framework for integration of variety of special-

purpose algorithms

semantic foundation– amazingly clean and elegant languages– roots in logic programming

applications– proven success

© Roman Barták, 1999

LimitationsNP-hard problems & tracktabilityunpredictable behaviourmodel stabilitytoo high-level

(new constraints, solvers, heuristics)

too low-level (modelling)

too localnon-incremental (rescheduling)

weak solver collaboration

© Roman Barták, 1999

Trendsmodelling

– global constraints (all_different)– modelling languages (Numerica, VisOpt)

understanding search– visualisation, performance debugging

hybrid algorithmssolver collaborationparallelismmulti-agent technology

© Roman Barták, 1999

Quotations“Constraint programming represents one of the

closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.”

Eugene C. Freuder, Constraints, April 1997

“Were you to ask me which programming paradigm is likely to gain most in commercial significance over the next 5 years I’d have to pick Constraint Logic Programming, even though it’s perhaps currently one of the least known and understood.”

Dick Pountain, BYTE, February 1995

© Roman Barták, 1999

ResourcesConferences

– Principles and Practice of Constraint Programming (CP)

– The Practical Application of Constraint Technologies and Logic Programming (PACLP)

Journal– Constraints (Kluwer Academic Publishers)

Internet– Constraints Archive

http://www.cs.unh.edu/ccc/archive

– Guide to Constraint Programminghttp://kti.mff.cuni.cz/~bartak/constraints/