© J. Christopher Beck 20081 Lecture 13: Modeling in Constraint Programming.

© J. Christopher Beck 2008 1

Lecture 13:Modeling inConstraint Programming


Outline Introduction

The Crystal Maze Heuristics, Propagation, B&B

Modeling in CP


Readings

P Ch 5.5, D.2, D.3

B.M. Smith, “Modelling”, The Handbook of Constraint Programming, 2006.


Crystal Maze Place the numbers 1 through 8 in the

nodes such that: Each number appears exactly once

?

?

?

?

?

?

??

– No connected nodes have consecutive numbers

You have 5 minutes!


Modeling

Each node a variable {1, …, 8} values in the domain of

each variable No consecutive numbers a

constraint (vi, vj) |vi – vj| > 1

All values used all-different constraint


Heuristic Search

?

?

?

?

?

?

?? 1 8

{1, 2, 3, 4, 5, 6, 7, 8}


Inference/Propagation

?

?

?

?

?

?

?? 1 8

{1, 2, 3, 4, 5, 6, 7, 8}

{1, 2, 3, 4, 5, 6, 7, 8}


Inference/Propagation

?

?

?

?

?

?

?? 1 8

{3, 4, 5, 6}

7 2

{3, 4, 5, 6}

{3, 4, 5, 6} {3, 4, 5, 6}

64

3 5

{3, 4, 5, 6, 7} {2, 3, 4, 5, 6}


Generic CP Algorithm

AssertCommitment

Propagators

Start

SuccessSolution?

MakeHeuristicDecision

BacktrackTechnique

Failure

Nothing toretract?

Dead-end?


With CP You Are …

Guaranteed to find a solution if one exists If not you can prove there is no solution

But you need to have enough time! On average, for many problems you can

find a solution within a reasonable time CP is commercially successful for

scheduling and other applications

Questions?


The Core of CP

Modeling How to represent the problem

Heuristic search How to branch How much effort to find a good branch

Inference/propagation How much effort

Backtracking


Constraint Satisfaction Problem (CSP)

Given: V, a set of variables {v0, v1, …, vn} D, a set of domains {D0, D1, …, Dn} C, a set of constraints {c0, c1, …, cm}

Each constraint, ci, has a scope ci(v0, v2, v4, v117, …), the variables that it constrains



A constraint, ci, is a mapping from the elements of the Cartesian product of the domains of the variables in its scope to {T,F} ci(v0, v2, v4, v117, …) maps:

(D0 X D2 X D4 X D117 X … ) {T,F}

A constraint is satisfied if the assignment of the variables in its scope are mapped to T



In a solution to a CSP: each variable is assigned a value

from its domain: vi = di, di є Di

each constraint is satisfied


Variables & Constraints

Variables can be anything colors, times, countries, … often integer valued or binary

Constraints are not limited to linear constraints extensional: list of acceptable value-

combinations intensional: mathematical expression


Examples

V = {v1, v2, …, vn} Di = {1, 2, …, n} all-diff(v1, v2, …, vn)

each of the variables must take on a different value

v1 > v2 + v4

v4 = v5 * v6


n-Queens Problem

Place n-Queens on the chess board so that no queen can attack any other


n-Queens Problem

Variables? Constraints?


n-Queens Problem

How about a different model?


Why Do We Care About Modeling?

You need to represent your problem!

As in MIP/LP different models are harder or easier to solve

We will return to CP modeling (in a big way) in a few weeks

© J. Christopher Beck 20081 Lecture 13: Modeling in Constraint Programming.

Documents

Transcript of © J. Christopher Beck 20081 Lecture 13: Modeling in Constraint Programming.