Constraint Programming Marco Kuhlmann & Guido Tack Lecture 1 - - PowerPoint PPT Presentation

Constraint Programming

Marco Kuhlmann & Guido Tack Lecture 1

Welcome!

Where am I?

• Constraint Programming
• advanced course
• 6 credit points
• lecture (2 hours) + lab (2 hours)
• WWW: http://www.ps.uni-sb.de/

This lecture

• constraint programming
• what it is – fundamental concepts
• why it matters – applications
• how it works – showcase examples
• organisation

Constraint Programming

S E N D M O R E + M O N E Y

Send More Money

Generate & Test

• generate

all possible combinations

• f all possible values

for all letters in the puzzle

• test

for each potential solution, whether it satisfies the equation

Ridiculous!

Specialised algorithms

• advantages
• may be highly efficient
• offer deep insights into the problem
• disadvantages
• may take years to develop
• cannot be easily adapted

Enter Constraint Programming

Constraint programming is a problem-solving technique for combinatorial problems that works by incorporating constraints into a programming environment. (after Apt 2003)

Combinatorial problems

• combinatorial structures

characterised in terms of a finite set of variables and a finite value for each variable

• combinatorial problem

question about properties

• f a class of combinatorial structures

subsets

Example

• combinatorial structures:

permutations of the primes between 1 and 10

• combinatorial problems:

How many such structures are there … without any additional properties? … where seq[0] > seq[1]? … where seq[0] + seq[1] = seq[2]?

Subsets & constraints

seq[0] > seq[1] seq[0] + seq[1] = seq[2]

subsets

Combinatorial problems

• combinatorial structures

characterised in terms of a finite set of variables and a finite value for each variable

• combinatorial problem

question about properties

• f a class of combinatorial structures

constraints

Constraint Satisfaction Problems (CSPs) simple formal model for combinatorial problems

CSPs: Ingredients

• finite set of problem variables, x
• associated domains dom(x)
• finite set of constraints

intensional

• r extensional

CSPs: Terminology

• variable assignment:

total function that maps each x to an element in dom(x)

• solution to a CSP:

variable assignment such that all constraints are satisfied

How to integrate constraints into a programming environment?

Meet Alice

• Alice is a better Standard ML:

it supports concurrent and distributed programming.

• Alice features the GeCoDE library,

the bleeding edge of research in constraint systems.

• WWW: http://www.ps.uni-sb.de/alice/

fun example1 space = let val dom = domain [2, 3, 5, 7] val sequence = fdtermVec (space, 4, dom) val seq0 = Vector.sub (sequence, 0) val seq1 = Vector.sub (sequence, 1) val seq2 = Vector.sub (sequence, 2) in distinct (space, sequence); (* seq[0] > seq[1] *) post (space, seq0 `> seq1); (* seq[0] + seq[1] = seq[2] *) post (space, seq0 `+ seq1 `= seq2); branch (space, sequence) end

Constraints in Alice

Constraint Propagation

Example

problem variables: x, y domains:

• dom(x) = {3,4,5},
• dom(y) = {3,4,5}

constraints:

• x ≥ y,
• y > 3

x = 4, y = 4 x = 5, y = 4 x = 5, y = 5 finite sets

• f integers

distinguish two sorts of constraints: basic constraints and non-basic constraints

x {3,4,5} y {3,4,5}

Constraint Store

basic constraints

Propagators

• implement non-basic constraints
• translate into basic constraints
• subscribe to variables in the store
• get notified about changes

Propagators

y > 3 x y

amplify store

y {3,4,5} x {3,4,5} y {4,5} x {4,5}

gets notified

x {3,4,5} y {3,4,5} x y y > 3

Computation Space

constraint store with connected propagators

Important concepts

• constraint store

stores basic constraints

• propagator

implements non-basic constraint

• computation space

constraint store + propagators

Branching

x {4,5} y {4,5} x y

Bad news

propagation is not enough!

Stable spaces

• solution

for each x, dom(x) is a singleton

• failure

there is an x with dom(x) empty

• choice

x {4,5} y {4,5} x y x {4} x {4} x {4} x {5} y {4,5} x y x y y {4,5} y {4}

Branching

stable space domain split

Search tree

choice solution

Branching heuristics

• naive heuristic
• pick some x with dom(x) > 1
• pick value k from dom(x)
• branch with x ∈ {n} and x ∉ {n}
• first-fail heuristic:

pick x with dom(x) minimal

Branching strategy

• variable selection
• any variable
• minimal/maximal current domain
• value selection (for the left branch)
• maximal/minimal/medial element
• lower half/upper half of the domain

Homework

Show that the branching strategy can have a massive impact

• n the size of the search tree.

Search

Search

• propagation and branching

induce a search tree

• orthogonal issue:

in what order are the nodes of that tree constructed?

• different problems require

different search strategies

Static search strategies

• explore the search tree
• standard search strategies
• depth-first search
• iterative deepening
• A* search

Dynamic search

• add new constraints during search
• dynamic search strategies
• iterative best-solution search
• branch-and-bound search

best solution search

Best Solution Search

• class of combinatorial structures σ
• objective function obj : σ → N
• find a structure s such that obj(s) is
• ptimal among all structures σ

Best Solution Search

• naive approach:

compute all solutions & choose best

• branch-and-bound approach:

compute first solution, s add ‘better-than-s’ constraint compute next solution, and iterate

prunes the search tree

S E N D M O S T + M O N E Y

Send Most Money

MONEY should be maximal

SMM+ – B & B

unexplored subtree add betterness constraint add betterness constraint best solution first solution

SMM+

SMM+

What this course will be about

Architecture

• propagation:

prune impossible values

• branching:

divide the problem into smaller parts

• search:

interleave propagation and branching to find solutions

What you will learn

• how to model combinatorial problems
• how to solve them using CP
• how to write new propagators
• how to program new search strategies
• how to apply CP to practical problems

Applications

• timetabling
• crew rostering
• gate allocation at airports
• sports league scheduling
• natural language processing

Scheduling resources

• tasks

duration, used resources

• precedence constraints

task a must precede task b

• resource constraints

at most one task per resource

First assignment

• install the tools:

http://www.ps.uni-sb.de/alice/

• warm-up in programming Alice
• model and solve some simple

constraint satisfaction problems

Constraint Programming

• can be used to tackle hard

combinatorial problems

• combines various interesting

methodologies and techniques

• applications are ubiquitous

knowledgeable people are not!

Constraint Programming

• compute with possible values

lower bound, upper bound

• prune inconsistent values

guessing as last resort

• factorise the problem

inferences + heuristics + search

What CP is not

• no efficiency miracle
• exponential problems remain

exponential

• If you have polynomial algorithms,

use them, and forget about CP!

• no replacement, but a complement

to other programming paradigms

What you should bring

• broad interest in computer science
• theory and formal models
• practice and programming
• self-initiative
• try! explore! do!
• ask questions, and answer them

Caveat

• CP is established …
• international conferences
• many results & applications
• … our tools are not (yet)
• some might not work (as expected)
• some might be uncomfortable

Organisation

Literature

• Krzysztof R. Apt:

Principles of Constraint Programming Cambridge University Press, 2003

• Christian Schulte:

Programming Constraint Services Springer-Verlag, 2002

Lectures

• 12 lectures in total
• last lecture on July 14
• no lectures on
• May 5 (Ascension)
• May 26 (Corpus Christi)

Labs

• explorative labs
• get familiar with the concepts
• get familiar with the tools
• graded labs
• four medium-sized projects
• determine 40% of your final grade

Exam

• at the end of the term
• determines 60% of final grade
• written or oral (to be determined)
• re-exam will be oral

Contact & Support

• mailing list

subscribe on web site

• office hours

Wednesdays, 14–15, room 45.517

• during & after the lectures

Constraint Programming

• problem-solving technique
• interleaves inferences & heuristics
• combines various methodologies
• is fun!

Thanks for your attention!

Backup Slides

Eight Queens

Problem specification

Place 8 queens on a chessboard such that no two queens attack each other.

History of Eight Queens

• originally proposed in 1842

by Max Bazzel, a chess player

• studied by various mathematicians
• George Pólya even studied it
• n donut-shaped chessboards

Model

• problem variables: row[0], … , row[7]
• dom(row[k]) = {0,…,7}
• assign row[k] = i iff

the queen in the k-th column should be placed in the i-th row

Constraints

• no two queens in the same column:

by construction

• no two queens in the same row:

distinct(row[0], … , row[7])

• no two queens in the same diagonal:

row[i] + (j - i) ≠ row[j] and row[i] – (j - i) ≠ row[j], for all 0 ≤ i < j ≤ 7

fun queens space = let val row = Linear.fdtermVec (space, 8, [0`#7]) in Linear.distinct (space, row); List.app (fn (i, j) => let val rowi = Vector.sub (row, i) val rowj = Vector.sub (row, j) in post (space, rowi `+ (`j `- `i) `<> rowj); post (space, rowi `– (`j `- `i) `<> rowj) end) (triangle 8); Linear.branch (space, row, FD.B_SIZE_MIN, FD.B_MED); row end

Eight Queens in Alice

Homework

Generalise the Eight Queens problem so that n queens are placed on an n-times-n chessboard.