Global Constraints

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

2. Course outline • Introduction • All Different • Lex ordering • Value precedence • Complexity • GAC-Schema • Soft Global Constraints • Global Grammar Constraints • Roots Constraint • Range Constraint • Slide Constraint • Global Constraints on Sets

3. SLIDE meta-constraint • Even hotter off the press than the value PRECEDENCE constraint • Under review for IJCAI 07!

4. SLIDE meta-constraint • A constructor for generating many sequencing and related global constraints • REGULAR • CONTIGUITY • LEX • CARD PATH • … • Slides a constraint down one or more sequences of variables • Ensuring constraint holds at every point • Fixed parameter tractable

5. Basic SLIDE • SLIDE(C,[X1,..Xn]) holds iff • C(Xi,..Xi+k) holds for every i • AMONG SEQ constraint • Used by ILOG for assembly line car sequencing at Renault • At most 1 in 3 cars have a sun roof • SLIDE(C,[X1,..Xn]) where C(X1,X2,X3) holds iff AMONG([X1,X2,X3],0,1,D) where D is set of cars ordered with sunroofs

6. GSC • Global sequence constraint in ILOG Solver • Combines AMONG SEQ with GCC • Regin and Puget give partial propagator • We can show why!

7. GSC • Global sequence constraint in ILOG Solver • Combines AMONG SEQ with GCC • Regin and Puget give partial propagator • We can show why! • It is NP-hard to enforce GAC on GSC • Can actually prove this when GSC is AMONG SEQ plus an ALL DIFFERENT • ALL DIFFERENT is a special case of GCC

8. GSC • Reduction from 1in3 SAT on positive clauses • The jth block of 2N clauses will ensure jth clause • Even numbered CSP vars represent truth assignment • Odd numbered CSP vars “junk” to ensure N odd values in each block • X_2jN+2i odd iff xi true • AMONG SEQ([X1,…],N,N,2N,{1,3,..}) • This ensures truth assignment repeated along variables!

9. GSC • Reduction from 1in3 SAT on positive clauses • Suppose jth clause is (x or y or z) • X_2jN+2x, X_2jN+2y, X_2jN+2z in {4NM+4j, 4NM+4j+1, 4NM+4j+2} • As ALL DIFFERENT, only one of these odd • For i other than x, y or z, X_2jN+2i in {4jN+4i, 4jN+4i+1} and X_2jN+2i+1 in {4jN+4i+2, 4jn+4i+3}

10. SLIDE down multiple sequences • Can SLIDE down more than one sequence at a time • LEX([X1,..Xn],[Y1,..Yn]) • Introduce sequence of Boolean vars [B1,..Bn+1] • Play role of alpha in LEX propagator

11. SLIDE down multiple sequences • Can SLIDE down more than one sequence at a time • LEX([X1,..Xn],[Y1,..Yn]) • Set B1=0, and Bn+1=1 (strict lex), Bn+1 in {0,1} (lex) • SLIDE(C,[X1..Xn],[Y1,..Yn],[B1,..Bn+1]) holds iff C(Xi,Yi,Bi,Bi+1) holds for each i

12. SLIDE down multiple sequences • Can SLIDE down more than one sequence at a time • LEX([X1,..Xn],[Y1,..Yn]) • Set B1=0, and Bn+1=1 (strict lex), Bn+1 in {0,1} (lex) • SLIDE(C,[X1..Xn],[Y1,..Yn],[B1,..Bn+1]) • C(Xi,Yi,Bi,Bi+1) holds iff Bi=1 or (Bi=Bi+1=0 and Xi=Yi) or (Bi=0, Bi+1=1 and Xi<Yi)

13. SLIDE down multiple sequences • Can SLIDE down more than one sequence at a time • LEX([X1,..Xn],[Y1,..Yn]) • Set B1=0, and Bn+1=1 (strict lex), Bn+1 in {0,1} (lex) • SLIDE(C,[X1..Xn],[Y1,..Yn],[B1,..Bn+1]) • C(Xi,Yi,Bi,Bi+1) holds iff Bi=1 or (Bi=Bi+1=0 and Xi=Yi) or (Bi=0, Bi+1=1 and Xi<Yi) • Highly efficient, incremental, ..

14. SLIDE down multiple sequences • CONTIGUITY([X1,..Xn]) • 0…01…10…0 • Two simple SLIDEs

15. SLIDE down multiple sequences • CONTIGUITY([X1,..Xn]) • 0…01…10…0 • Two simple SLIDEs • Introduce Yi in {0,1,2} • SLIDE(>=,[Y1,..Yn])

16. SLIDE down multiple sequences • CONTIGUITY([X1,..Xn]) • 0…01…10…0 • Two simple SLIDEs • Introduce Yi in {0,1,2} • SLIDE(>=,[Y1,..Yn]) • SLIDE(C,[X1,..Xn],[Y1,..Yn]) where C(Xi,Yi) holds iff Xi=1 <-> Yi=1

17. SLIDE down multiple sequences • REGULAR(Q,[X1,..Xn]) • X1 .. Xn is a string accepted by FDA Q • Encodes into simple SLIDE • Introduce Yi to represent state of the automaton after i symbols

18. SLIDE down multiple sequences • REGULAR(Q,[X1,..Xn]) • X1 .. Xn is a string accepted by FDA Q • Introduce Yi to represent state of the automaton after i symbols • SLIDE(C,[X1,..Xn],[Y1,..Yn+1]) where • Y1 is starting state of Q • Yn+1 is limited to accepting states of Q • C(Xi,Yi,Yi+1) holds iff Q moves from state Yi to state Yi+1 on seeing Xi

19. SLIDE down multiple sequences • REGULAR(A,[X1,..Xn]) • X1 .. Xn is a string accepted by FDA A • Introduce Qi to represent state of the automaton after i symbols • SLIDE(C,[X1,..Xn],[Q1,..Qn+1]) where • Y1 is starting state of A • Yn+1 is limited to accepting states of A • C(Xi,Qi,Qi+1) holds iff A moves from state Qi to state Qi+1 on seeing Xi • Gives highly efficient and effective propagator!

20. SLIDE with counters • AMONG([X1,..Xn],v,N) • Introduce sequence of counts, Yi • SLIDE(C,[X1,..Xn],[Y1,..Yn+1]) where • Y1=0, Yn+1=N • C(Xi,Yi,Yi+1) holds iff (Xi in v and Yi+1=1+Yi) or (Xi not in v and Yi+1=Yi)

21. SLIDE with counters • CARD PATH • SLIDE is a special case of CARD PATH • SLIDE(C,[X1,..Xn]) iff CARD PATH(C,[X1,..Xn],n-k+1)

22. SLIDE with counters • CARD PATH • SLIDE is a special case of CARD PATH • SLIDE(C,[X1,..Xn]) iff CARD PATH(C,[X1,..Xn],n-k+1) • CARD PATH is a special case of SLIDE • SLIDE(D,[X1,..Xn],[Y1,..Yn+1]) where Y1=0, Yn+1=N, and D(Xi,..Xi+k,Yi,Yi+1) holds iff (Yi+1=1+Yi and C(Xi,..Xi+k)) or (Yi+1=Yi and not C(Xi,..Xi+k))

23. SLIDE with parameters • Slide constraints may share parameters • LINKSET2BOOLEANS(S,[X1,..Xn]) • Converts set variable into characteristic function • Encodes as SLIDE(C,[X1,..Xn]) where • C(S,Xi) holds iff Xi in S • S is parameter common to each slide constraint

24. SLIDE over sets • Value precedence for set vars • PRECENDCE([vj,vk],[S1,..Sn]) holds iff • min(i,{i | vj in Si and vk not in Si or i=n+1}) < • min(i,{i | vk in Si and vj not in Si or i=n+2})

25. SLIDE over sets • Value precedence for set vars • PRECENDCE([vj,vk],[S1,..Sn]) holds iff • min(i,{i | vj in Si and vk not in Si or i=n+1}) < • min(i,{i | vk in Si and vj not in Si or i=n+2}) • Introduce sequence of Booleans to indicate whether vars have been distinguished apart yet or not

26. SLIDE over sets • Value precedence for set vars • PRECENDCE([vj,vk],[S1,..Sn]) holds iff • min(i,{i | vj in Si and vk not in Si or i=n+1}) < • min(i,{i | vk in Si and vj not in Si or i=n+2}) • SLIDE(C,[S1,..Sn],[B1,..Bn+1]) where • B1=0 and • C(Si,Bi,Bi+1) holds iff Bi=Bi+1=1, or Bi=Bi+1=0 and (vj, vk in Si or vj, vk not in Si), or Bi=0, Bi+1=1, vj in Si and vk not in Si

27. SLIDE over sets • Open stacks problem • IJCAI 05 modelling challenge • Three SLIDEs and one ALL DIFFERENT • First SLIDE: Si+1 = Si u customer(Xi) • Second SLIDE: Ti-1= Ti u customer(Xi) • Third SLIDE: |Si intersect Ti| < OpenStacks

28. Circular SLIDE • STRETCH used in shift rostering • Given sequence of vars X1,.. Xn • Each stretch of identical values a occurs at least shortest(a) and at most longest(a) time • For example, at least 0 and at most 3 night shifts in a row • Each transition Xi=/=Xi+1 is limited to given patterns • For example, only Xi=night, Xi+1=off is permitted

29. Circular SLIDE • STRETCH can be efficiently encoded using SLIDE • SLIDE(C,[X1,..Xn],[Y1,..Yn+1]) where • Y1=1 • C(Xi,Xi+1,Yi,Yi+1) holds iff Xi=Xi+1, Yi+1=1+Yi, Yi+1<=longest(Xi), or Xi=/=Xi+1, Yi>=shortest(Xi) and (Xi,Xi+1) in set of permitted changes

30. Circular SLIDE • Circular forms of STRETCH are needed for repeating shift patterns • Circular form of SLIDE useful in such situations • SLIDEo(C,[X1,..Xn]) holds iff • C(Xi,..X1+(i+k-1)mod n) holds for 1<=i<=n

31. SLIDE algebra • SLIDEOR(C,[X1..Xn]) holds iff • C(Xi,..Xi+k) holds for some I • Encodes as CARD PATH (and thus as SLIDE) • Other more complex combinations • NOT(SLIDE(C,[X1,..Xn])) iff SLIDEOR(C,[X1,..Xn]) • SLIDE(C1,[X1,..Xn]) and SLIDE(C2,[X1,..Xn]) iff SLIDE(C1 and C2,[X1,..Xn]) • ..

32. Propagating SLIDE • But how do we propagate global constraints expressed using SLIDE?

33. Propagating SLIDE • SLIDE(C,[X1,..Xn]) • Just post sequence of constraints, C(Xi,..Xi+k) • If constraint graph is Berge acyclic, then we will achieve GAC • Gives efficient GAC propagators for CONTIGUITY, DOMAIN, ELEMENT, LEX, PRECEDENCE, REGULAR, …

34. Propagating SLIDE • SLIDE(C,[X1,..Xn]) • Just post sequence of constraints, C(Xi,..Xi+k) • If constraint graph is Berge acyclic, then we will achieve GAC • Gives efficient GAC propagators for CONTIGUITY, DOMAIN, ELEMENT, LEX, PRECEDENCE, REGULAR, … • But what about case constraint graph is not Berge-acyclic? • Slide constraints overlap on more than one variable

35. Propagating SLIDE • SLIDE(C,[X1,..Xn]) • Slide constraints overlap on more than one variable • Enforce GAC using dynamic programming • pass support down sequence

36. Propagating SLIDE • SLIDE(C,[X1,..Xn]) • Equivalently a “dual” encoding • Consider AMONG SEQ(2,2,3,[X1,..X5],{a}) where • X1=a, X2,X3,X4,X5 in {a,b} • AMONG([X1,X2,X3],2,{a}) • AMONG([X2,X3,X4],2,{a}) • AMONG([X3,X4,X5],2,{a}) • Enforcing GAC sets X4=a • GAC can be enforced in O(nd^k+1) time and O(nd^k) space where constraints overlap on k variables • Fixed parameter tractable

37. Conclusions • SLIDE is a very useful meta-constraint • Many global constraints for sequencing and other problems can be encoded as SLIDE • SLIDE can be propagated easily • Constraints overlap on just one variable => simply post slide constraints • Constraints overlap on more than one variable => use dynamic programming or equivalently a simple dual encoding