Global Constraints and Constraint Programming

1. Global Constraints and Constraint Programming Michael Maher Loyola University Chicago

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

3. 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).

4. 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.

5. 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.

6. 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.

7. 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

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

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

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

11. 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.

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

13. 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.

14. 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.

15. Improve on AC • Stronger consistency conditions

16. Improve on AC • Stronger consistency conditions • Weaker consistency conditions

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

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

19. 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

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