Constraint Programming: What is behind?

Constraint Programming:What is behind?

Roman Barták

Charles University, [email protected]

© 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


Talk ScheduleHistorical contextConstraint technology

– basic features– constraint satisfaction– constraints in optimisation– over-constrained problems

ApplicationsSummary

– Advantages & Limitations

TrendsResources


The Origins

Artificial Intelligence– Scene Labelling (Waltz)

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

Logic Programming– unification --> constraint solving

Operations Research– NP-hard combinatorial problems


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

+

+

+ -

+

- ++ +

+


Interactive GraphicsSketchpad (Sutherland)ThingLab (Borning)

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


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


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)


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


Systematic Search Methods

exploring the solution spacecomplete and soundefficiency issues

Backtracking (BT)Generate & Test (GT)

exploringindividual assignments

exploring subspace

X

Z

Y


Generate & Test

probably the most general problem solving method

Algorithm:generate labelling

test satisfaction

Drawbacks: Improvements:blind generator smart generator

--> local search

late discovery of testing within generatorinconsistencies --> backtracking



Backtracking (BT)incrementally extends a partial solution

towards a complete solutionAlgorithm:

assign value to variable

check consistency

until all variables labelled

Drawbacks:– thrashing– redundant work– late detection of conflict


A

C

B

1

1

1

1

2

2

A = D, B D, A+C < 4

1

2

D21

1


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



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


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


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


X

Y

Z

X

Y

Z


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

1 2

1 2

1 2


X

YZ


Path Consistency (PC)

consistency along the path only

checking paths of length 2 is enoughPlus/Minus

+ detects more inconsistencies than AC

- extensional representation of constraints

- changes in graph connectivity

Directional PC, Restricted PC


V0 V1

V3

V2 V4 V5

???


K -consistency

K-consistency– consistent valuation o (K-1) variables can be

extended to K-th variable

strong K-consistency J-consistency for each JK

NC strong 1-consistencyAC strong 2-consistencyPC strong 3-consistency



Consistency Completenessstrongly N-consistent constraint graph with

N nodes => solutionstrongly K-consistent constraint graph with

N nodes (K<N) => ???

path consistentbut no solution

Special graph structures – tree structured graph => (D)AC is enough– cycle cutset, MACE

A

B C

D

{1,2,3}

{1,2,3}

{1,2,3}

{1,2,3}


Constraint Propagationsystematic search only => no efficientconsistency only => no completecombination of search (backtracking) with

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

look back

Labelling order

look ahead


Look Back Methodsintelligent backtrackingconsistency checks among instantiated

variables

backjumping– backtracks to the conflicting variable

backchecking and backmarking– avoids redundant constraint checking

by remembering conflicting levelfor each value

Constraint Propagation

jump here

a

b b b

conflict

still conflict


Look Ahead Methodspreventing future conflicts via consistency

checks among not yet instantiated variables

forward checking (FC)– AC to direct neighbourhood

partial look ahead (PLA)– DAC

(full) look ahead (LA)– Arc Consistency– Path Consistency

Constraint Propagationlab

elling

ord

er

instantiated variable


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


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 neighbouring configuration randomly– tabu search

few last configurations are forbidden for next step

does not guarantee completeness


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


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


Branch & Bounddepth first search (like BT)

+ under estimate of the objective function (minimisation)

+ bound (initially set to plus infinity)

heuristic function: partial labelling -> under estimate of the objective function

pruning sub-tree under the partial labelling when– constraint inconsistency detected

– heuristic value exceeds the bound

efficiency is determined by:– the quality of the heuristic function

– whether a good bound is found early

Constraint Optimisation


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


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


Partial CSPweakening a problem:

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

one solutionwhite - denim - sneakers


enlarged constraint’s domain

red white

blue denim grey

cordovans sneakers

shirt

trousers footwear


Partial CSPPartial Constraint Satisfaction Problem– CSP– evaluation function: labelling -> numerical value

Task:find labelling optimal regarding the evaluation functionExample: maximising number of satisfied constraints

Usage:– over-constrained problems– optimisation problems (PCSP is a generalisation of CSOP)– obtaining “good enough” solution within fixed time



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


red white

blue denim grey

cordovans sneakers

shirt

trousers footwear


Constraint HierarchiesHierarchy = (finite) set of labelled constraints

levels Hi (H0 - required constraints, H1 - strongest non required …)

Solution:– S0={ | all required constraints are satisfied by }

– SH={ | S0 s.t. S0 better(,,H) }where better - comparator (partial ordering of valuations)

solving methods:– refining method (DeltaStar)

• solve constraints from stronger to weaker levels

– local propagation (DeltaBlue, SkyBlue)• repeatedly selects uniquely satisfiable constraints• planning + value propagation

hierarchies in optimisation problems– weak constraint objective function = optimum



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


Scheduling Problemsthe most successful application area

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


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


LimitationsNP-hard problems & tracktabilityunpredictable behaviourmodel stabilitytoo high-level

(new constraints, solvers, heuristics)

too low-level (modelling)

too localnon-incremental (rescheduling)

weak solver collaboration


Trendsmodelling

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

understanding search– visualisation, performance debugging

hybrid algorithmssolver collaborationparallelismmulti-agent technology


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/

Constraint Programming: What is behind?

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