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Global Constraints and Constraint Programming Michael Maher Loyola University Chicago

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Constraint Personalities Type A aggressive, time-sensitive Type B relaxed, deliberate There are two kinds of constraints:

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Type A Constraints Composition of constraints is totally understood by a constraint solver Intervals in Solver, CHIP, … Domains in CSP Linear inequalities in CLP(R) Term equations in Prolog Linear inequalities in Math Prog These constraints can be solved algorithmically (at low cost).

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Constraint Solver Constraint store holding all active Type A constraints. On-line algorithms for deciding consistency, implication, … Constraint store changes by addition/undo/deletion of constraints.

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Type B Constraints Composition of constraints is at best partly understood Global constraints Arithmetic constraints in Solver, CHIP, … Constraints in CSP Integrality in Math Prog These are the reason we do search.

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In FD solvers, arithmetic inequalities are Type B constraints. From x + y ≤ z, z ≤ y + 2 the system does not understand x ≤ 2. Intervals are Type A constraints. From x :: 1..5 and x :: 3..7 the system understands x :: 3..5.

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Type B constraints are implemented as propagators of Type A constraints x :: 1..3, y :: 2..6 x + y ≤ z z ≤ y + 2 z :: 3.. z :: 3..8

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Global Constraints Collections of many small constraints that are treated as a unit alldifferent cumulative cycle etc Not global !

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Global Constraints Implementations of global constraints are encapsulations of on-line algorithms alldifferent cumulative cycle etc Often (?) the implementations are based on CO techniques.

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Constraint Programming is a software architecture that supports the combination of CO algorithms to solve problems.

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Arc Consistency Ideally, an implementation will propagate all information expressible with Type A constraints that is a consequence of the Type B constraint. If B & c → c’ then c → c’ This is a general form of arc consistency.

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Arc Consistency The definition covers many existing definitions: Arc consistency in CSP Interval consistency in FD Echidna consistency Hull consistency Rule consistency

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Arc Consistency Existing forms of arc consistency only admit unary Type A constraints. They are all weakenings of arc consistency in CSP. We can also use other Type A constraints (like ordering x ≤ y) to get different, not weaker, consistencies.

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Minimum min(x, y, z) For AC we need the following propagations, among others: min(x, y, z) → z ≤ x, z ≤ y x ≤ y, min(x, y, z) → z = x u ≤ x, u ≤ y, min(x, y, z) → u ≤ z Implementing non-unary AC is hard.

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Improve on AC Stronger consistency conditions

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Improve on AC Stronger consistency conditions Weaker consistency conditions

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Improve on AC Stronger consistency conditions consider more than one global constraint at a time Weaker consistency conditions

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Improve on AC Stronger consistency conditions consider more than one global constraint at a time Weaker consistency conditions avoid or delay complete propagation

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Exercises for COs Define and implement new global constraints Define and implement constraint solver for new class of constraints Introduce us to the Dark Arts

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Exercises for BRs Formulate and implement BR in a conjunctive constraint context Extend BR to situations where tests for consistency are not complete.

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