1. Non-binary constraints Toby Walsh Cork Constraint Computation Center tw@4c.ucc.ie

2. Global constraints Toby Walsh Cork Constraint Computation Center tw@4c.ucc.ie

3. Non-exhaustive catalog Order constraints Constraints on values Partitioning constraints Timetabling constraints Graph constraints Scheduling constraints Bin-packing constraints

4. Global constraints It isn’t just all-different! Many constraints specialized to application domains  Scheduling  Packing ..

5. Order constraints min(X,[Y1,..,Yn]) and max(X,[Y1,..Yn]) X <= minimum(Y1,..,Yn) X >= maximum(Y1,..Yn) n.b. min is relational not functional

6. Order constraints min(X,[Y1,..,Yn]) and max(X,[Y1,..Yn]) X <= minimum(Y1,..,Yn) X <= maximum(Y1,..Yn) n.b. min is relational not functional min_mod(X,[Y1,..,Yn],m) and max_mod(X,[Y1,..Yn],m) X mod m <= minimum(Y1 mod m,..,Yn mod m) X mod m <= maximum(Y1 mod m,..,Yn mod m)

7. Order constraints min_n(X,n,[Y1,..Ym]) and max_n(X,n,[Y1,..,Ym) X is nth smallest value in Y1,..Ym X is nth largest value in Y1,..Ym

8. Order constraints min_n(X,n,[Y1,..Ym]) and max_n(X,n,[Y1,..,Ym) X is nth smallest value in Y1,..Ym X is nth largest value in Y1,..Ym min_n(X,1,[Y1,..,Ym]) => min(X,[Y1,..,Yn]) max_n(X,1,[Y1,..,Ym]) => max(X,[Y1,..,Ym]) n.b. min_n is functional but min is relational (any X smaller than min will do!

9. Value constraints among(N,[Y1,..,Yn],[val1,..,valm]) N vars in [Y1,..,Yn] take values val1,..valm e.g. among(2,[1,2,1,3,1,5],[3,4,5])

10. Value constraints among(N,[Y1,..,Yn],[val1,..,valm]) N vars in [Y1,..,Yn] take values val1,..valm e.g. among(2,[1,2,1,3,1,5],[3,4,5]) count(n,[Y1,..,Ym],op,X) where op is =,,=/, = relation “Yi op X” holds n times E.g. among(n,[Y1,..,Ym],[k]) = count(n,[Y1,..,Ym],=,k)

11. Value constraints balance(N,[Y1,..,Yn]) N = occurrence of more frequent value - occurrence of least frequent value E.g balance(2,[1,1,1,3,4,2])

12. Value constraints balance(N,[Y1,..,Yn]) N = occurrence of more frequent value - occurrence of least frequent value E.g balance(2,[1,1,1,3,4,2]) all-different([Y1,..,Yn]) => balance(0,[Y1,..,Yn])

13. Value constraints min_nvalue(N,[Y1,..,Yn]) and max_nvalue(N,[Y1,..,Yn]) least (most) common value in Y1,..,Yn occurs N times E.g. min_nvalue(2,[1,1,2,2,2,3,3,5,5]) Can replace multiple count or among constraints

14. Value constraints common(X,Y,[X1,..,Xn],[Y1,..,Ym]) X vars in Xi take a value in Yi Y vars in Yi take a value in Xi E.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9])

15. Value constraints common(X,Y,[X1,..,Xn],[Y1,..,Ym]) X vars in Xi take a value in Yi Y vars in Yi take a value in Xi E.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9]) among(X,[Y1,..,Yn],[val1,..,valm]) = common(X,Y,[X1,..,Yn],[val1,..,valm])

16. Value constraints same([X1,..,Xn],[Y1,..,Yn]) Yi is a permutation of Xi

17. Value constraints same([X1,..,Xn],[Y1,..,Yn]) Yi is a permutation of Xi used_by([X1,..,Xn],[Y1,..,Ym]) all values in Yi are used by vars in Xi m>=m

18. Value constraints same([X1,..,Xn],[Y1,..,Yn]) Yi is a permutation of Xi used_by([X1,..,Xn],[Y1,..,Ym]) all values in Yi are used by vars in Xi m>=m on n values: alldifferent([X1,..,Xn])=same([X1,..,Xn],[1,..,n]) =used_by([X1,..,Xn],[1,..,n])

19. Partitioning constraints all-different([X1,..,Xn])

20. Partitioning constraints all-different([X1,..,Xn]) Other flavours all-different_except_0([X1,..,Xn]) Xi=/Xj unless Xi=Xj=0 0 is often used for modelling purposes as “dummy” value  Don’t use this slab  Don’t open this bin..

21. Partitioning constraints all-different([X1,..,Xn]) Other flavours symmetric-all-different([X1,..,Xn]) Xi=/Xj and Xi=j iff Xj=i Very common in practice  Team i plays j iff Team j plays i..

22. Partitioning constraints all-different([X1,..,Xn]) Other flavours symmetric-all-different([X1,..,Xn]) Xi=/Xj and Xi=j iff Xj=i Very common in practice  Team i plays j iff Team j plays i..  Regin has proposed very efficient algorithm

23. Partitioning constraints nvalue(N,[X1,..,Xn]) Xi takes N different values all-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn)

24. Partitioning constraints nvalue(N,[X1,..,Xn]) Xi takes N different values all-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn) gcc([X1,..,Xn],Lo,Hi) values in Xi occur between Lo and Hi times all-different([X1,..,Xn])=gcc([X1,..,Xn],1,1)

25. Timetabling constraints change(N,[X1,..,Xn]),op) where op is {=,, =,/=} “Xi op Xi+1” holds N times E.g. change(3,[4,4,3,4,1],/=) You may wish to limit the number of changes of classroom, shifts, …

26. Timetabling constraints longest_changes(N,[X1,..,Xn]),op) where op is {=,, =,/=} longest sequence “Xi op Xi+1” is of length N E.g. longest_changes(2,[4,4,4,3,3,2,4,1,1,1],=) You may wish to limit the length of a shift without break, …

27. Graph constraints Tours in graph a often represented by the successors: [X1,..,Xn] means from node i we go to node Xi

28. Graph constraints Tours in graph a often represented by the successors: [X1,..,Xn] means from node i we go to node Xi E.g. [2,1,5,3,4] represents the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3)

29. Graph constraints cycle(N,[X1,..,Xn]) there are N cycles in Xi e.g. cycle(2,[2,1,5,3,4]) as we have the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) Useful for TSP like problems (e.g. sending engineers out to repair phones)

30. Graph constraints derangement(N,[X1,..,Xn]) there are no length 1 cycles in Xi

31. Graph constraints derangement(N,[X1,..,Xn]) there are no length 1 cycles in Xi e.g. derangement([2,1,5,3,4]) as the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) have length 2 and 3

32. Graph constraints derangement(N,[X1,..,Xn]) there are no length 1 cycles in Xi e.g. derangement([2,1,5,3,4]) as the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) have length 2 and 3 e.g. derangement([3,1,2,4]) as the 2 cycles (1)->(3)->(2)- >(1) and (4)->(4) are of length 3 and 1

33. Graph constraints We may also wish to represent predecessors  For good labelling heuristics  For ease of statement of constraints [X1,..,Xn] means we came to node i from node Xi

34. Graph constraints We may also wish to represent predecessors  For good labelling heuristics  For ease of statement of constraints [X1,..,Xn] means we came to node i from node Xi E.g. [2,1,4,5,3] now represents the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3)

35. Graph constraints We may also wish to represent predecessors  For good labelling heuristics  For ease of statement of constraints [X1,..,Xn] means we came to node i from node Xi E.g. [2,1,4,5,3] now represents the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) In successor notation this was [2,1,5,3,4]

36. Graph constraints We may also wish to represent predecessors  For good labelling heuristics  For ease of statement of constraints [X1,..,Xn] means we came to node i from node Xi E.g. [2,1,4,5,3] now represents the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) In successor notation this was [2,1,5,3,4]  How do we know they are the same?

37. Graph constraints We may also wish to represent predecessors  For good labelling heuristics  For ease of statement of constraints [X1,..,Xn] means we came to node i from node Xi E.g. [2,1,4,5,3] now represents the 2 cycles (1)->(2)->(1) and (3)->(5)->(4)->(3) In successor notation this was [2,1,5,3,4]  How do we know they are the same?  With a channelling constraint!

38. Graph constraints Inverse([X1,..,Xn],[Y1,..,Yn]) Where [X1,..,Xn] is a successor representation of a path, and [Y1,..,Yn] is a predecessor Thus inverse([2,1,4,5,3],[2,1,5,3,4])

39. Graph constraints Inverse([X1,..,Xn],[Y1,..,Yn]) Where [X1,..,Xn] is a successor representation of a path, and [Y1,..,Yn] is a predecessor Thus inverse([2,1,4,5,3],[2,1,5,3,4]) And inverse([2,1,5,3,4],[2,1,4,5,3])

40. Scheduling constraints cummulative([S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],L) schedules n (concurrent) jobs, each with a height Hi ith job starts at Si, runs for Di and ends at Ei  Ei=Si+Di at any time, accumulated height of running jobs is less than L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

41. Scheduling constraints coloured_cummulative([S1,..,Sn],[D1,..,Dn],[E1,..,E n],[C1,..,Cn],L) schedules n (concurrent) each with a colour Ci no more than L colours running at any one time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

42. Scheduling constraints cycle_cummulative(m,[S1,..,Sn],[D1,..,Dn],[E1,.., En],[H1,..,Hn],L) schedules n (concurrent) jobs, each with a height Hi onto a cyclic schedule of length m

43. Scheduling constraints cummulatives([M1,,,Mn],[S1,..,Sn],[D1,..,Dn],[E 1,..,En],[H1,..,Hn],[L1,..,Lm]) schedules n (concurrent) jobs, each with a height Hi onto one of m machines ith runs on Mi accumulated height of running jobs on machine i <= Li

44. Scheduling constraints cummulatives([M1,..Mn],[S1,..,Sn],[D1,..,Dn],[E1,..,En],[H1,..,Hn],[L1,..,Lm]) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Machine 1 Machine 2

45. Scheduling constraints coloured_cummulatives([M1,,,Mn],[S1,..,Sn],[D1,..,Dn],[E1,..,En],[C1,..,Cn],[L1,..,Lm]) schedules n (concurrent) jobs, each with a colour i onto one of m machines ith runs on Mi number of colours of running jobs on machine i <= Li

46. Bin-packing constraints bin_packing(capacity,[B1,..,Bn],[w1,..,wn]) for each bin j, sum_Bi=j wi <= capacity

47. Bin-packing constraints bin_packing(capacity,[B1,..,Bn],[w1,..,wn]) for each bin j, sum_Bi=j wi <= capacity special case of cummulative with task durations=1

48. Bin-packing constraints bin_packing(capacity,[B1,..,Bn],[w1,..,wn]) for each bin j, sum_Bi=j wi <= capacity special case of cummulative with task durations=1 unlike normal bin-packing, does not minimize the number of open bins

49. Bin-packing constraints bin_packing(capacity,[B1,..,Bn],[w1,..,wn]) for each bin j, sum_Bi=j wi <= capacity special case of cummulative with task durations=1! unlike normal bin-packing, does not minimize the number of open bins how could we do this?

50. Bin-packing constraints bin_packing(capacity,[B1,..,Bn],[w1,..,wn]) for each bin j, sum_Bi=j wi <= capacity special case of cummulative with task durations=1! unlike normal bin-packing, does not minimize the number of open bins Bi=j implies Open_j=1 Minimize sum_j Open_j

51. Misc constraints element(Index,[a1,..,an],Var) Var=a_Index constraint programming’s answer to arrays! e.g. element(Item,[10,23,12,15],Cost)

52. GAC algorithms

53. GAC-Schema A generic framework for achieving AC for any kind of constraint (can be non binary). Bessiere and Regin, IJCAI’97 You just have to say how to compute a solution. Works incrementality (notion of support).

54. GAC-Schema: instantiation List of allowed tuples List of forbidden tuples Predicates Any OR algorithm Solver re-entrace

55. GAC-Schema Idea: tuple = solution of the constraint support = valid tuple - while the tuple is valid: do nothing - if the tuple is no longer valid, then search for a new support for the values it contains a solution (support) can be computed by any algorithm

56. Example X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)}

57. Example X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)} Support for (x1,a): (a,a,a) is computed and (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a) in (a,a,a) is marked as supported.

58. Example X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)} Support for (x1,a): (a,a,a) is computed and (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a) in (a,a,a) is marked as supported. Support for (x2,a): (a,a,a) is in S(x2,a) it is valid, therefore it is a support. (Multidirectionnality). No need to compute a solution

59. Example X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)} Support for (x1,a): (a,a,a) is computed and (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a) in (a,a,a) is marked as supported. Value a is removed from x1, then all the tuples in S(x1,a) are no longer valid: (a,a,a) for instance. The validity of the values supported by this tuple must be reconsidered.

60. Example X(C)={x1,x2,x3} D(xi)={a,b} T(C)={(a,a,a),(a,b,b),(b,b,a),(b,b,b)} Support for (x1,a): (a,a,a) is computed and (a,a,a) is added to S(x2,a) and S(x3,a), (x1,a) in (a,a,a) is marked as supported. Support for (x1,b): (b,b,a) is computed, and updated...

61. GAC-Schema: complexity CC complexity to check consistency (seek in table, call to OR algorithm): seek for a Support costs CC n variables, d values: for each value: CC for all values: O(ndCC) For any OR algorithm which is able to compute a solution, Arc consistency can be achieved in O(ndCC).

62. Table Constraint: An example Configuration problem: 5 types of components: {glass, plastic, steel, wood, copper} 3 types of bins: {red, blue, green} whose capacity is red 5, blue 5, green 6 Constraints: - red can contain glass, cooper, wood - blue can contain glass, steel, cooper - green can contain plastic, copper, wood - wood require plastic; glass exclusive copper - red contains at most 1 of wood - green contains at most 2 of wood For all the bins there is either no plastic or at least 2 plastic Given an initial supply of 12 of glass, 10 of plastic, 8 of steel, 12 of wood and 8 of copper; what is the minimum total number of bins?

63. Table Constraint: results #bk time standard model 1,361,709 430 Table Constraint 12,659 9.7

64. Semantics of a constraint Speed-up the search for a support (solution which contain a value (x,a))

65. Exploiting constraint semantics Speed-up the search for a support (solution which contain a value (x,a)): x < y, D(x)=[0..10000], D(y)=[0..10000] support for (x,9000): immediate any value greater than 9000 in D(y)

66. Semantics of a constraint Design of ad-hoc filtering algorithm: x < y : (a)max(x) = max(y) -1 (b)min(y) = min(x) +1

67. Exploiting semantics Design of ad-hoc filtering algorithm: x < y : (a)max(x) = max(y) -1 (b)min(y) = min(x) +1 Triggering of the filtering algorithm: no possible pruning of D(x) while max(y) is not modified no possible pruning of D(y) while min(x) is not modified

68. Regin’s alldiff algorithm Construct the value graph

69. x1 x2 x3 x4 x5 x6 12345671234567 The value graph: D(x1)={1,2} D(x2)={2,3} D(x3)={1,3} D(x4)={3,4} D(x5)={2,4,5,6} D(x6)={5,6,7}

70. Alldiff constraint Construct a flow network

71. Flow network x1 x2 x3 x4 x5 x6 12345671234567 s Default orientation

72. Flow network x1 x2 x3 x4 x5 x6 12345671234567 ts Default orientation

73. Flow network x1 x2 x3 x4 x5 x6 12345671234567 ts Default orientation

74. Flow network x1 x2 x3 x4 x5 x6 12345671234567 ts (6,6) (0,1) (1,1) Default orientation

75. Alldiff constraint Construct a flow network Compute a feasible flow

76. Flow network x1 x2 x3 x4 x5 x6 12345671234567 ts (6,6) (0,1) (1,1) Default orientation

77. A feasible flow x1 x2 x3 x4 x5 x6 12345671234567 ts (6,6) (0,1) (1,1) Default orientation

78. Alldiff constraint Construct a flow network Compute a feasible flow Construct the residual graph  Reverse direction of arcs on feasible flow (that is, of red arcs)

79. A feasible flow x1 x2 x3 x4 x5 x6 12345671234567 ts (6,6) (0,1) (1,1) Default orientation

80. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

81. Alldiff constraint Construct a flow network Compute a feasible flow Construct the residual graph Compute the strongly connected components  Strongly connected component = maximal subgraph such that if directed path u to v, also from v to u

82. Alldiff constraint Construct a flow network Compute a feasible flow Construct the residual graph Compute the strongly connected components  Strongly connected component = maximal subgraph such that if directed path u to v, also from v to u  Exists O(N) algorithm due to Tarjan (1972) = O(n+d)

83. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

84. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

85. Alldiff constraint Construct a flow network Compute a feasible flow Construct the residual graph Compute the strongly connected components  Remove arcs of flow value 0 (i.e. black arcs) with ends belonging to two different components

86. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

87. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

88. GAC for all-different Theorem An arc is deleted from residual graph iff the associated value is not GAC

89. Residual graph x1 x2 x3 x4 x5 x6 12345671234567 s orientation

90. GAC subproblem x1 x2 x3 x4 x5 x6 12345671234567 The value graph: D(x1)={1,2} D(x2)={2,3} D(x3)={1,3} D(x4)={4} D(x5)={5,6} D(x6)={5,6,7}

91. Before! x1 x2 x3 x4 x5 x6 12345671234567 The value graph: D(x1)={1,2} D(x2)={2,3} D(x3)={1,3} D(x4)={3,4} D(x5)={2,4,5,6} D(x6)={5,6,7}

92. Conclusions Vast range of global constraints  Cover all the areas that CP cover  Some very generic (e.g. all-different)  Some very specific (coloured cummulative..) GAC algorithms  Generic (GAC-schema)  Specific (GAC alldifferent)