Constraint Satisfaction Problems: Lecture Notes - Prof. Introduction to Artificial Intelli, Summaries of Material Engineering

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Set 5: Constraint Satisfaction Problems

Chapter 6 R&N

ICS 271 Fall 2016 Kalev Kask

Outline

• The constraint network model
  • Variables, domains, constraints, constraint graph, solutions
• Examples:
  • graph-coloring, 8-queen, cryptarithmetic, crossword puzzles, vision problems,scheduling, design
• The search space and naive backtracking,
• The constraint graph
• Consistency enforcing algorithms
  • arc-consistency, AC-1,AC- 3
• Backtracking strategies
  • Forward-checking, dynamic variable orderings
• Special case: solving tree problems
• Local search for CSPs

A B

red green red yellow green red green yellow yellow green yellow red

Constraint Satisfaction

Example: map coloring

Variables - countries (A,B,C,etc.)

Values - colors (e.g., red, green, yellow)

Constraints: A  B , A  D , D  E , e tc.

C

A

B

D

E

F

G

Hard vs Soft Constraints

• Hard constraints : must be satisfied
  • Satisfaction problem
• Soft constraints : capture preferences
  • Optimization problem

Sudoku

Each row, column and major block must be all different

“Well posed” if it has unique solution: 27 constraints

Example : The 4-queen problem

Q

Q

Q

Q Q

Q

Q

Q

Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.

Standard CSP formulation of the problem:

• Variables : each row is a variable.

Q

Q

Q

Q

X 1

X 4

X 3

X 2

• Domains : Di { 1 , 2 , 3 , 4 }.
• Constraints : There are ( 42 ) = 6 constraints involved: R 12 {( 1 , 3 )( 1 , 4 )( 2 , 4 )( 3 , 1 )( 4 , 1 )( 4 , 2 )} R 13 {( 1 , 2 )( 1 , 4 )( 2 , 1 )( 2 , 3 )( 3 , 2 )( 3 , 4 )( 4 , 1 )( 4 , 3 )} R 14 {( 1 , 2 )( 1 , 3 )( 2 , 1 )( 2 , 3 )( 2 , 4 )( 3 , 1 )( 3 , 2 )( 3 , 4 )( 4 , 2 )( 4 , 3 )} R 23 {( 1 , 3 )( 1 , 4 )( 2 , 4 )( 3 , 1 )( 4 , 1 )( 4 , 2 )} R 24 {( 1 , 2 )( 1 , 4 )( 2 , 1 )( 2 , 3 )( 3 , 2 )( 3 , 4 )( 4 , 1 )( 4 , 3 )} R 34 {( 1 , 3 )( 1 , 4 )( 2 , 4 )( 3 , 1 )( 4 , 1 )( 4 , 2 )} - Constraint Graph : X 1

X (^2) X 4

X 3

Class scheduling/Timetabling

But only dn^ unique assignments