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Constraint Processing and Programming Introductory Exemple Javier Larrosa

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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

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Place numbers 1 through 8 on nodes each number appears exactly once no connected nodes have consecutive numbers ? ? ? ? ? ? ?? Note the graph symmetry Acknowledgement: Patrick Prosser

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Heuristic Search Which nodes are hardest to number? ? ? ? ? ? ? ?? Guess a value, but be prepared to backtrack

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Heuristic Search ? ? ? ? ? ? ?? Which nodes are hardest to number?

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Heuristic Search Which are the least constraining values to use? ? ? ? ? ? ? ?? Values 1 & 8

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

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Inference/propagation We can now eliminate many values for other nodes ? 1 ? ? 8 ? ?? {1,2,3,4,5,6,7,8}

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Inference/propagation By symmetry ? 1 ? ? 8 ? ?? {3,4,5,6}

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Inference/propagation ? 1 ? ? 8 ? ?? {3,4,5,6} {1,2,3,4,5,6,7,8}

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Inference/propagation By symmetry ? 1 ? ? 8 ? ?? {3,4,5,6}

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Inference/propagation ? 1 ? ? 8 ? ?? {3,4,5,6} {3,4,5,6,7}{2,3,4,5,6}

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Inference/propagation ? 1 ? ? 8 ? 27 {3,4,5,6}

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Inference/propagation ? 1 ? ? 8 ? 27 {3,4,5,6} And propagate

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Inference/propagation ? 1 ? ? 8 ? 27 {3,4,5} {4,5,6}

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Inference/propagation 3 1 ? ? 8 ? 27 {3,4,5} {4,5,6} By symmetry

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Inference/propagation 3 1 ? ? 8 ? 27 {3,4,5} {4,5,6} And propagate

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Inference/propagation 3 1 ? ? 8 ? 27 {4,5} {5,6} {4,5,6} More propagation?

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Inference/propagation 3 1 4 5 8 6 27 A solution

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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.

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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

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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 Satisfaction Problem

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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

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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

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Constraint model for puzzle variables v 1, …, v 8 domains {1, …, 8} constraints | v 1 – v 2 | 1 | v 1 – v 3 | 1 … | v 7 – v 8 | 1 alldifferent(v 1, …, v 8 ) ? ? ? ? ? ? ??

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Example: instruction scheduling (a + b) + c Given a basic-block of code and a single-issue pipelined processor, find the minimum schedule

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Example: evaluate (a + b) + c instructions A r1 a B r2 b C r3 c D r1 r1 + r2 E r1 r1 + r3 33 3 1 AB DC E dependency DAG

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Example: evaluate (a + b) + c non-optimal schedule Ar1 a Br2 b nop Dr1 r1 + r2 Cr3 c nop Er1 r1 + r3 33 3 1 AB DC E dependency DAG

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Example: evaluate (a + b) + c optimal schedule Ar1 a Br2 b Cr3 c nop Dr1 r1 + r2 Er1 r1 + r3 33 3 1 AB DC E dependency DAG

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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) 33 3 1 AB DC E dependency DAG

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

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Example: 3-coloring variables: v 1, v 2, v 3, v 4, v 5 domains: {1, 2, 3} constraints: v i v j if v i and v j are adjacent v2v2 v3v3 v1v1 v5v5 v4v4

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Example: 3-coloring A solution v 1 1 v 2 2 v 3 2 v 4 1 v 5 3 v2v2 v3v3 v1v1 v5v5 v4v4

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

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Constraint model 4 3 2 1 x1 x1 x2x2 x3 x3 x4 x4 variables: x 1, x 2, x 3, x 4 domains: {1, 2, 3, 4} constraints: x i x j and | x i - x j | | i - j |

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Example: 4-queens A solution x 1 2 x 2 4 x 3 1 x 4 3 Q Q Q Q x1 x1 x2x2 x3 x3 x4 x4 4 3 2 1

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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

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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

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Application areas Paradigm of choice for many hard combinatorial problems scheduling planning vehicle routing configuration bioinformatics …

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Commercial applications

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Constraint Satisfaction Problems

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