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Global Constraints

Global Constraints . Toby Walsh NICTA and University of New South Wales www.cse.unsw.edu.au/~tw. Overview: a tale of 3 global constraints. Value PRECEDENCE Symmetry breaking Decomposition does not hinder propagation NVALUES Computational complexity to the rescue REGULAR

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Global Constraints

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  1. Global Constraints Toby Walsh NICTA and University of New South Wales www.cse.unsw.edu.au/~tw

  2. Overview: a tale of 3 global constraints • Value PRECEDENCE • Symmetry breaking • Decomposition does not hinder propagation • NVALUES • Computational complexity to the rescue • REGULAR • Formal languages to the rescue

  3. Why GLOBAL constraints? • The one thing that people in LP most envy about CP! • Capture common patterns • E.g. AllDifferent: variables take all different values • Each queen must be on a different column

  4. Why GLOBAL constraints? • The one thing that people in LP most envy about CP! • Capture common patterns • Provide effective pruning • E.g. complex pigeonhole arguments (these 4 variables have just 4 values between them so other variables must take values outside)

  5. Why GLOBAL constraints? • The one thing that people in LP most envy about CP! • Capture common patterns • Provide effective pruning • E.g. complex pigeonhole arguments (these 4 variables have just 4 values between them so other variables must take values outside) • Do so efficiently • E.g. O(n log n) time to enforce BC on AllDifferent

  6. Typical CP solver • Backtracking search • Try Queen1=col1 • Try Queen2=col3 • Else Queen2=col4 • Else … • Else Queen1=col2 • Try Queen2=col4 • Else Queen2=col5 • … • Else Queen1=col3 • …

  7. Typical CP solver • Backtracking search • Try Queen1=col1 • Constraint pruning removes • Queen2=col1 • Queen2=col2 • Queen3=col1 • Queen3=col3 • …

  8. Search tree • Branching • Queen1=col1 • Queen2=col3 • .. • Propagation AllDifferent(Queen1,..,Queen8)

  9. Propagators • Effective pruning • Enforce GAC or BC • Complex pigeonhole arguments • X1, X3, X6  {1,2,3} then X2  {1,2,3} • Efficiently • BC on AllDifferent in O(n log n) time • GAC on AllDifferent on O(n2.5) time • Called each node in a potentially exponential sized search tree!

  10. Propagators • Incremental • Each time we call them only a little changes (i.e. one variable is instantiated) • On backtracking, need to restore their state quickly (efficient data structures to do so like watched literals)

  11. Propagation • GAC • Each value for each variable can be extended to a satisfying assignment (aka support) of the global constraint • E.g. AllDifferent({1,2},{1,2},{1,2,3})

  12. Propagation • GAC • Each value for each variable can be extended to a satisfying assignment (aka support) of the global constraint • AC • GAC on binary constraints

  13. Propagation • GAC • Each value for each variable can be extended to a satisfying assignment (aka support) of the global constraint • BC • Max and min can be extended to a bound support (assignment between max and min)

  14. Global constraint • Constraint over parameterized number of variables • E.g. AllDifferent(X1,..,Xn) • n is not fixed • Compare binary constraint • X < Y • Fixed to 2 variables

  15. Global constraint • Constraint over parameterized number of variables • E.g. AllDifferent(X1,..,Xn) • n is not fixed • Cannot use brute force to make GAC • dn possible supports to try out in general • Must use smart algorithms!

  16. Value PRECEDENCE Global constraint for dealing with symmetry in a problem

  17. Value precedence • Used to deal with value symmetry • Prevent us finding symmetric (equivalent) solutions • Good example of “global” constraint where we can use an efficient encoding • Encoding gives us GAC • Asymptotically optimal, achieve GAC in O(nd) time • Good incremental complexity

  18. Value symmetry • Decision variables: • Col[Italy], Col[France], Col[Austria] ... • Domain of values: • red, yellow, green, ... • Constraints Col[Italy]=/=Col[France] Col[Italy]=/=Col[Austria] …

  19. Value symmetry • Solution: • Col[Italy]=green Col[France]=red Col[Spain]=green … • Values (colours) are interchangeable: • Swap red with green everywhere will still give us a solution

  20. Value symmetry • Solution: • Col[Italy]=red Col[France]=green Col[Spain]=red … • Values (colours) are interchangeable: • Swap red with green everywhere will still give us a solution

  21. Value precedence • Old idea • Used in bin-packing and graph colouring algorithms • Only open the next new bin • Only use one new colour • Applied now to constraint satisfaction

  22. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1

  23. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1 • For X2, we need only consider two choices • X2=1 or X2=2

  24. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1 • For X2, we need only consider two choices • Suppose we try X2=2

  25. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1 • For X2, we need only consider two choices • Suppose we try X2=2 • For X3, we need only consider three choices • X3=1, X3=2, X3=3

  26. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1 • For X2, we need only consider two choices • Suppose we try X2=2 • For X3, we need only consider three choices • Suppose we try X3=2

  27. Value precedence • Suppose all values from 1 to m are interchangeable • Might as well let X1=1 • For X2, we need only consider two choices • Suppose we try X2=2 • For X3, we need only consider three choices • Suppose we try X3=2 • For X4, we need only consider three choices • X4=1, X4=2, X4=3

  28. Value precedence • Global constraint • Precedence([X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) for all j<k • In other words • The first time we use j is before the first time we use k • Now cannot swap j with k so symmetry is broken

  29. Value precedence • Global constraint • Precedence([X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) • E.g • Precedence([1,1,2,1,3,2,4,2,3]) • But not Precedence([1,1,2,1,4])

  30. Value precedence • Global constraint • Precedence([X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) • Propagator proposed by [Law and Lee 2004] • Pointer based propagator but only for two interchangeable values at a time

  31. Value precedence • Precedence([j,k],[X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) • Of course • Precedence([X1,..Xn]) iff Precedence([i,j],[X1,..Xn]) for all i<j

  32. Value precedence • Precedence([j,k],[X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) • Of course • Precedence([X1,..Xn]) iff Precedence([i,i+1],[X1,..Xn]) for all i

  33. Value precedence • Precedence([j,k],[X1,..Xn]) iff min({i | Xi=j or i=n+1}) < min({i | Xi=k or i=n+2}) • But this hinders propagation • GAC(Precedence([X1,..Xn])) does strictly more pruning than GAC(Precedence([i,j],[X1,..Xn])) for all i<j • Consider X1=1, X2 in {1,2}, X3 in {1,3} and X4 in {3,4}

  34. Puget’s method • Introduce Zj to record first time we use j • Add constraints • Xi=j implies Zj <= i • Zj=i implies Xi=j • Zi < Zi+1

  35. Puget’s method • Introduce Zj to record first time we use j • Add constraints • Xi=j implies Zj <= i • Zj=i implies Xi=j • Zi < Zi+1 • Binary constraints • easy to implement

  36. Puget’s method • Introduce Zj to record first time we use j • Add constraints • Xi=j implies Zj <= I • Zj=i implies Xi=j • Zi < Zi+1 • Unfortunately hinders propagation • AC on encoding may not give GAC on Precedence([X1,..Xn]) • Consider X1=1, X2 in {1,2}, X3 in {1,3}, X4 in {3,4}, X5=2, X6=3, X7=4

  37. Propagating Precedence • Simple ternary encoding • Introduce Yi+1 for largest value used up to Xi • Post sequence of constraints C(Xi,Yi,Yi+1) for each 1<=i<=n These hold iff Xi<=1+Yi and Yi+1=max(Yi,Xi)

  38. Propagating Precedence • Post sequence of constraints • C(Xi,Yi,Yi+1) for each 1<=i<=n • These hold iff Xi<=1+Yi and Yi+1=max(Yi,Xi) • Consider Y1=0, X1 in {1,2,3}, X2 in {1,2,3} and X3=3

  39. Propagating Precedence • Post sequence of constraints • C(Xi,Yi,Yi+1) for each 1<=i<=n • These hold iff Xi<=1+Yi and Yi+1=max(Yi,Xi) • Decomposition does not hinder propagation • Enforcing GAC on decomposition makes Precedence([X1,…,Xn]) GAC • Reason: constraint graph is Berge-acyclic

  40. Berge acyclicity • Constraint graph • Node for each variable • Edge between variables in the same constraint • Berge acyclicity • Graph does not contain any cycles (tree) • Local consistency = global consistency Trees are our friends

  41. Partial value precedence • Values may partition into equivalence classes • Values within each equivalence class are interchangeable • E.g. Shift1=nursePaul, Shift2=nursePeter, Shift3=nurseJane, Shift4=nursePaul ..

  42. Partial value precedence • Shift1=nursePaul, Shift2=nursePeter, Shift3=nurseJane, Shift4=nursePaul .. • If Paul and Jane have the same skills, we can swap them (but not with Peter who is less qualified) • Shift1=nurseJane, Shift2=nursePeter, Shift3=nursePaul, Shift4=nurseJane …

  43. Partial value precedence • Values may partition into equivalence classes • Value precedence easily generalized to cover this case • Within each equivalence class, vi occurs before vj for all i<j (ignore values from other equivalence classes)

  44. Partial value precedence • Values may partition into equivalence classes • Value precedence easily generalized to cover this case • Within each equivalence class, vi occurs before vj for all i<j (ignore values from other equivalence classes) • For example • Suppose vi are one equivalence class, and ui another

  45. Partial value precedence • Values may partition into equivalence classes • Value precedence easily generalized to cover this case • Within each equivalence class, vi occurs before vj for all i<j (ignore values from other equivalence classes) • For example • Suppose vi are one equivalence class, and ui another • X1=v1, X2=u1, X3=v2, X4=v1, X5=u2

  46. Partial value precedence • Values may partition into equivalence classes • Value precedence easily generalized to cover this case • Within each equivalence class, vi occurs before vj for all i<j (ignore values from other equivalence classes) • For example • Suppose vi are one equivalence class, and ui another • X1=v1, X2=u1, X3=v2, X4=v1, X5=u2 • Since v1, v2, v1 … and u1, u2, …

  47. Conclusions • Symmetry of interchangeable values can be broken with value precedence constraints • Value precedence can be decomposed into ternary constraints • Efficient and effective method to propagate • Berge acyclic constraint graph • Can be generalized in many directions • Partial interchangeability, …

  48. NVALUES Computational complexity to the rescue!

  49. Global constraints • Hardcore algorithms • Data structures • Graph theory • Flow theory • Combinatorics • … • Computational complexity • Global constraints are often balanced on the limits of tractability!

  50. Computational complexity 101 • Some problems are essentially easy • Multiplication, O(n1.58) • Sorting, O(n log n) • Regular language membership, O(n) • Context free language membership, O(n3) .. • P (for “polynomial”) • Class of decision problems recognized by deterministic Turing Machine in polynomial number of steps • Decision problem • Question with yes/no answer? E.g. is this string in the regular language? Is this list sorted? …

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