Tutorial Constraint Satisfaction Problems. CSP Exercise 1

Tutorial

Constraint Satisfaction Problems

CSP

Exercise 1

• Consider the following map. The task is to color the map using the three colors Red, Blue, and Green, such that no two adjacent regions take the same color.

1. Formulate the problem as a CSP. Clearly state the variables, domains, and constraints.

2. Describe the topology of the constraint graph.

3. What is the size of the search space?

4. Color the map using:

1. Backtracking Search with MRV heuristic. 2. Backtracking Search with MCV heuristic. 3. Backtracking Search with LCV heuristic. 4. Backtracking Search with Forward Checking heuristic.

5. Note: Variables and values are ordered as follows:

• Variables: WA, NT, Q, NSW, V, SA, T • Values: Red, Blue, and Green

1. Backtracking Search with MRV heuristic.

2. Backtracking Search with MCV heuristic.

3. Backtracking Search with LCV heuristic.

4. Backtracking Search with Forward Checking.

CSP

Exercise 2

Use all heuristics including arc-propagation to solve this problem.

CE

D

B

F

A

G

H

{blue, green, red} {blue, green, red}

{blue, green, red}

{blue, green, red}

{blue, green, red}

{blue, green, red}

{blue, green, red}

{blue, green, red}

3 colour me!

3 colour me using Backtracking and Forward Checking …

CE

D

B

F

A

G

H

{blue, green, red} { , green, red}

{blue, , }

{ , green, }

{blue, , red}

{ , , red}

{blue, , }

{ , green, }

3 colour me!

CSP

Exercise 3

CSP – Examination Problem

• During an examination, an invigilator has to arrange six students in a way to make sure that the examination is properly conducted and no one cheats. 2 students have makeup in Artificial Intelligence, 1 student has makeup in Database, 2 students have makeup in Mathematics, and 1 student in Physics. As shown in figure below, there exist 6 desks arranged into two rows and three columns.

Examination Problem• The problem for the invigilator is to arrange students so that

students having the same makeup exam should not be neighbors (in the same column, in the same row or the same diagonal for example - neighbors of 5 are 1, 2, 3, 4 and 6, - and neighbors of 3 are 2 (same row), 5 (same diagonal) and 6 (same column)).

1. Propose a formulation of the problem in term of CSP by specifying variables, domains and constraints

2. Describe the topology of the constraint graph.

3. Using Forward Checking with Backpropagation, propose a solution to help the invigilator. Variables and values should be selected according to alphabetical or ascending order.

4. Using Backpropagation with MCV heuristic, propose a solution to help the invigilator. Variables and values are ordered according to alphabetical or ascending order.

Examination Problem - Solution

1. Using Forward Checking with Backpropagation, propose a solution to help the invigilator. Variables and values should be selected according to alphabetical or ascending order.

2. Using Backpropagation with MCV heuristic, propose a solution to help the invigilator. Variables and values are ordered according to alphabetical or ascending order.

AI DB AI

M P M

CSP

Exercise 4

Backtracking + FC

Forward Checking

• Do it yourself …

• Solve 6-queens …

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Q3Q2 Q4Q1 Q5

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Q6

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 6 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 5 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 6 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 5 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 6 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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6

Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 5 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 5 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

1

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32 41 5

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6

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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6 Column 6 has domain wipeoutBacktrack to next “choice”

Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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Forward Checking

• Solve 6-queens …

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32 41 5

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First Solution!