Constraint Processing and ProgrammingIntroductory Exemple

Javier Larrosa

Combinatorial Problem Solving

• Science and Engineering are full of Combinatorial Problems

Solving algorithms are exponentially expensive A lot of tricks can be applied to improve

performance

• Naive Approach

For each problem, design and implement an efficient algorithm

Place numbers 1 through 8 on nodes

• each number appears exactly once

• no connected nodes have consecutive numbers

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Note the graph symmetry

Acknowledgement: Patrick Prosser

Heuristic Search

Which nodes are hardest to number?

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Guess a value, but be prepared to backtrack

Heuristic Search

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Which nodes are hardest to number?

Heuristic Search

Which are the least constraining values to use?

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Values 1 & 8

Heuristic Search

Symmetry means we don’t need to consider: 8 1

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Inference/propagation

We can now eliminate many values for other nodes

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

Inference/propagation

By symmetry

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{3,4,5,6}

{3,4,5,6}

Inference/propagation

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{3,4,5,6}

{3,4,5,6}

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

Inference/propagation

By symmetry

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8

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{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

Inference/propagation

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8

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{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

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

Inference/propagation

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8

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{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

Inference/propagation

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8

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{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

{3,4,5,6}

And propagate

Inference/propagation

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27

{3,4,5}

{3,4,5}

{4,5,6}

{4,5,6}

Inference/propagation

3

1

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8

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27

{3,4,5}

{4,5,6}

{4,5,6}

By symmetry

Inference/propagation

3

1

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8

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27

{3,4,5}

{4,5,6}

{4,5,6}

And propagate

Inference/propagation

3

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{4,5}

{5,6}

{4,5,6}

More propagation?

Inference/propagation

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8

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27

A solution

The Declarative Approach to Combinatorial Problem Solving

• Declarative Approach

1. Model the problem using a well-defined language 2. Solve the problem using general-purpose

techniques

Constraint Programming and SAT solving follow this approach.

Constraint programming methodology

• Model problem

• Solve model

• Verify and analyze solution

• specify in terms of constraints on acceptable solutions• define/choose constraint model:

variables, domains, constraints

• define/choose algorithm• define/choose heuristics

Constraint programming methodology

• Model problem

• Solve model

• Verify and analyze solution

• specify in terms of constraints on acceptable solutions• define/choose constraint model:

variables, domains, constraints

• define/choose algorithm• define/choose heuristics

ConstraintSatisfaction

Problem

Constraint satisfaction problem

• A CSP is defined by

• a set of variables

• a domain of values for each variable

• a set of constraints between variables

• A solution is

• an assignment of a value to each variable that satisfies the constraints

Given a CSP

• Determine whether it has a solution or not

• Find any solution

• Find all solutions

• Find an optimal solution, given some cost function

Constraint model for puzzle

variables

v1, …, v8

domains

{1, …, 8}

constraints

| v1 – v2 | 1

| v1 – v3 | 1

…

| v7 – v8 | 1

alldifferent(v1, …, v8)

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Example: instruction scheduling

(a + b) + cGiven a basic-block of code and a single-issue pipelined processor, find the minimum schedule

Example: evaluate (a + b) + c

instructions

A r1 a

B r2 b

C r3 c

D r1 r1 + r2

E r1 r1 + r3

3 3

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A B

D C

E

dependency DAG

Example: evaluate (a + b) + c

non-optimal schedule

A r1 a

B r2 b

nop

nop

D r1 r1 + r2

C r3 c

nop

nop

E r1 r1 + r3

3 3

31

A B

D C

E

dependency DAG

Example: evaluate (a + b) + c

optimal schedule

A r1 a

B r2 b

C r3 c

nop

D r1 r1 + r2

E r1 r1 + r3

3 3

31

A B

D C

E

dependency DAG

Constraint model

variables

A, B, C, D, E

domains

{1, …, m}

constraints

D A + 3

D B + 3

E C + 3

E D + 1

alldifferent(A, B, C, D, E)

3 3

31

A B

D C

E

dependency DAG

Example: Graph coloring

Given k colors, does there exist a coloring of the nodes such that adjacent nodes are assigned different colors

Example: 3-coloring

variables: v1, v2 , v3 , v4 , v5

domains: {1, 2, 3}

constraints: vi vj if vi and vj

are adjacent

v2

v3

v1

v5

v4

Example: 3-coloring

A solution

v1 1

v2 2

v3 2

v4 1

v5 3

v2

v3

v1

v5

v4

Example: n-queens

Place n-queens on an n n board so that no pair of queens attacks each other

Constraint model

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1

x1 x2 x3 x4variables: x1, x2 , x3 , x4

domains: {1, 2, 3, 4}

constraints: xi xj and

| xi - xj | | i - j |

Example: 4-queens

A solution

x1 2

x2 4

x3 1

x4 3

Q

Q

Q

Q

x1 x2 x3 x4

4

3

2

1

Constraint programming methodology

• Model problem

• Solve model

• Verify and analyze solution

• specify in terms of constraints on acceptable solutions• define/choose constraint model:

variables, domains, constraints

• define/choose algorithm• define/choose heuristics

Constraint programming methodology

• Model problem

• Solve model

• Verify and analyze solution

• specify in terms of constraints on acceptable solutions• define/choose constraint model:

variables, domains, constraints

• define/choose algorithm• define/choose heuristics

Application areas

• Paradigm of choice for many hard combinatorial problems

• scheduling

• planning

• vehicle routing

• configuration

• bioinformatics

• …

Commercial applications