Propagating Separable Equalities. in an MDD Store. Peter Tiedemann INFORMS John Hooker. Tarik Hadzic. Cork Constraint Computation Centre

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1 Propagating Separable Equalities in an MDD Store Tarik Hadzic Cork Constraint Computation Centre John Hooker Carnegie Mellon University Peter Tiedemann Configit Software Slide 1 INFORMS 2008

2 Slide 2 Constraint Programming Strength of CP processes individual constraints. Exploits problem substructure. Weakness of CP processes individual constraints. No global point of view. Overlooks implications of combined constraints. Two compensating strategies: Global constraints Domain store relaxation

3 The domain store Stores the current variable domains Values that occur in some feasible solution Provides a global point of view. to some extent. Pools results of individual constraint processing. Basis for constraint propagation. Slide 3

4 Advantages of domain store Provides natural input into filtering algorithms. Filters start with current domains when processing a constraint. Guides branching (on variables) in a natural way. Simply split the domain in the current domain store. Slide 4

5 Slide 5 Disadvantages of domain store Transmits relatively little information. A weak relaxation of the problem. Ignores interactions between variables. Feasible set is simply a Cartesian product of domains. Unbalanced tradeoff. Search trees too large. Too little processing at nodes.

6 A stronger discrete relaxation Relaxed multivalued decision diagram can serve as a constraint store. Generalization of binary decision diagram, used in circuit verification, configuration problems, etc. An MDD is a compact representation of a search tree for the problem. Isomorphic subtrees are merged. MDD is relaxed by limiting its width. Slide 6

7 Slide 7 Advantages of MDD store A much stronger relaxation than domain store. Strength of relaxation is adjustable, ranging from domain store to original problem. Guides branching in a natural way. More guidance than domain store. Justifies more processing at search tree nodes. Better integration of CP/IP.

8 Example problem x + x = 4 x 1 3 x 1 x 2 x = 3 3 x 1 + x 2 5 j { 1,2,3 } (,, ) (1,1,2) Slide 8

9 Search tree x x x Infeasible Feasible Slide 9

10 Remove infeasible solutions 1 2 x 1 3 x x Slide 10

11 Merge isomorphic subtrees 1 2 x 1 3 x 2 {2,3} 1 {1,2,3} x Slide 11 {1,2}

12 Remove redundant edge 1 2 x 1 3 x 2 {2,3} 1 {1,2,3} x Slide 12 {1,2}

13 Reduced MDD x x 2 {2,3} 1 x Slide 13 {1,2}

14 x + x = 4 x 1 3 x 1 x 2 x = 3 3 x 1 + x 2 5 j { 1,2,3 } (,, ) (1,1,2) x 1 Edge domain x 2 {2,3} x 3 Slide 14 u 1 u 2 u 3 u 4 u 5 u 6 1 {1,2}

15 Each path represents a cartesian product of solutions {2,3} Slide 15 x 1 x 2 {2,3} u 1 u 2 u 3 {1,2} x 3 u 4 u 5 u 6 1

16 Each path represents a cartesian product of solutions {1,2,3} Slide 16 x 1 x 2 {2,3} u 1 u 2 u 3 {1,2} x 3 u 4 u 5 u 6 1

17 MDD Relaxation Let s use relaxed MDD with max width of 2 Full MDD: 8 solutions Relaxed MDD: 14 solutions Slide 17 x 1 x 2 {2,3} u 1 u 2 u 3 {1,2} x 3 u 4 u 5 {1,2} 1

18 MDD Relaxation MDD relaxation with width 1 is just the domain store. Full MDD: 8 solutions Relaxed MDD: 14 solutions Domain store: = 27 solutions Slide 18 x 1 x 2 x 3 u 1 {1,2,3} u 2 {1,2,3} u 5 {1,2,3} 1

19 Branching search using relaxed MDD x 1 {1,2,3} x 2 {1,2,3} x 1 x 2 {2,3} u 1 u 2 u 3 {1,2} x 3 u 4 u 5 {1,2} Slide 19 1

20 Branching search using relaxed MDD x 1 {1,2,3} x 2 {1,2,3} {2,3} x 3 {1,2} x 1 x 2 x 3 u 1 u 2 u 3 {1,2} u 4 u 5 {1,2} Slide 20 1

21 Branching search using relaxed MDD x 1 {1,2,3} x 2 {1,2,3} {2,3} {2,3} x 3 {1,2} x 3 x 1 x 2 u 1 u 2 u 3 {1,2} x 3 u 4 u 5 {1,2} Slide 21 1

22 Branching search using relaxed MDD x 1 {1,2,3} {2,3} x 2 {1,2,3} x 2 {1,2,3} {2,3} x 3 {1,2} x 3 x 1 x 2 {2,3} u 1 u 2 u 3 {1,2} Slide 22 And so forth. Less branching than with domain store. x 3 u 4 u 5 1 {1,2}

23 Propagation in MDDs We can propagate a constraint in an MDD relaxation by edge domain filtering and refinement (node splitting). We take care not to exceed max width. Example: We will propagate alldiff in an MDD relaxation of width 3. Slide 23 Andersen, Hadzic, Hooker & Tiedemann, A constraint store based on multivalued decision diagrams, 2007

24 Propagation in MDDs Current MDD relaxation Slide 24 u 1 u 2 u 3 {1,2} u 4 {1,2,3} u 5 {1,2} u 6 u 6

25 Propagation in MDDs First filter edge domains using alldiff Slide 25 u 1 u 2 u 3 {1,2} u 4 {1,2,3} u 5 {1,2} u 6 u 6

26 Propagation in MDDs First filter edge domains using alldiff Slide 26 u 1 u 2 u 3 u 4 {1,2} u 5 {1,2} u 6 u 6

27 Propagation in MDDs To split u 5 : Identify equivalence classes of incoming edges Slide 27 u 1 u 2 u 3 u 4 {1,2} u 5 {1,2} u 6 u 6

28 Propagation in MDDs To split u 5 : Identify equivalence classes of incoming edges These are equivalent for alldiff because they lead to the same set of feasible paths. u 1 u 2 u 3 u 5 {1,2} u 4 Slide 28 u 6 u 6

29 Propagation in MDDs To split u 5 : Identify equivalence classes of incoming edges Slide 29 These are equivalent for alldiff because they lead to the same set of feasible paths. Easy to check equivalence for alldiff all incoming paths use same values u 1 u 2 u 3 {1,2} u 6 u 5 u 6 u 4

30 Propagation in MDDs To split u 5 : u 1 Identify equivalence classes of incoming edges. u 2 u 3 u 4 Split u 5 to receive 3 equivalence classes. u 5 u 5 u 5 {1,2} Slide 30 u 6 u 6

31 Propagation in MDDs Duplicate outgoing edges. Slide 31 u 1 u 2 u 3 u 4 u 5 u 5 u 5 {1,2} u 6 u 6

32 Propagation in MDDs Duplicate outgoing edges. u 1 u 2 u 3 u 4 u 5 u 5 u 5 {1,2} {1,2} Slide 32 u 6 {1,2} u 6

33 Propagation in MDDs Filter domains. u 1 u 2 u 3 u 4 u 5 u 5 u 5 {1,2} {1,2} Slide 33 u 6 {1,2} u 6

34 Propagation in MDDs Filter domains. u 1 u 2 u 3 u 4 u 5 u 5 u 5 Slide 34 u 6 u 6

35 Propagation in MDDs Alldiff has now been propagated. u 1 u 2 u 3 u 4 Hadzic, Hooker, O Sullivan & Tiedemann, Approximate compilation of constraints into multivalued decision diagrams, 2008 Slide 35 u 5 u 5 u 5 u 6 u 6

36 Slide 36 Computational results

37 Problems Multiple alldiffs. Three overlapping alldiffs variables alldiff(,, ) x 1 x n Separable equalities. Three nonlinear equalities in 15 variables. Variable domains contain 3 values. i g i ( x i ) = β Slide 37

38 Results Multiple alldiffs Conventional domain store About a million search tree nodes. MDD constraint store No backtracking. Even for width as small as 5. Wider MDDs require fewer splits. Slide 38

39 Slide 39 Number of branches + number of splits Domain store MDD-based Max width of MDD

40 Slide 40 Computation time (milliseconds) Domain store MDD-based Max width of MDD

41 MDDs are faster for all but smallest widths. Speed advantage of MDDs increases with maximum width. Up to a factor of about 30. Within the range of widths studied. Slide 41

42 Results Separable Equalities MDD propagation cannot compete with MILP for solving separable equalities alone. Slide 42

43 Results Separable Equalities MDD propagation cannot compete with MILP for solving separable equalities alone. MDD propagation is designed for problems that are suitable for CP. but contain some separable equalities. We want effective propagation of the equalities. Slide 43

44 Results Separable Equalities MDD propagation cannot compete with MILP for solving separable equalities alone. MDD propagation is designed for problems that are suitable for CP. but contain some separable equalities. We want effective propagation of the equalities. We compare MDD propagation for equalities with MDD propagation for 2 inequalities i.e., replacing each equality with 2 inequalities And propagating each inequality separately. Slide 44

45 Filter edge domains with dynamic programming. Applied separately for each equality. Use shortest and longest path lengths to the terminus to prune the calculation. Slide 45

46 Checking for edge equivalence is hard. Two partial solutions (x 1,, x k ) = (a 1,, a k ) (x 1,, x k ) = (b 1,, b k ) are equivalent iff k n g i ( a i ) + g i ( x i ) = β i = 1 i = k + 1 has same solutions as k n g i ( b i ) + g i ( x i ) = β i i k = 1 = + 1 Slide 46

47 Checking for edge equivalence is hard. Checking whether k k = g ( a ) g ( b ) i i i i i = 1 i = 1 is too strict Slide 47

48 Check for approximate equivalence. This is constraint-specific. Allows us to exploit special structure. In the case of equalities, check whether k k g ( a ) g ( b ) i i i i i = 1 i = 1 Slide 48

49 Actually, we did something else. Split on the nodes u i for which L max ( u i ) L min ( u i ) Longest path from u i to terminus Shortest path from u i to terminus Slide 49

50 MDD propagation of an equality is much faster than propagation of 2 inequalities. Both performance and advantage increase with MDD width. Slide 50

51 Slide 51 MDD propagation of inequalities MDD propagation of equalities

52 Slide 52 Conclusions and research issues

53 Conclusions MDD store provides substantial advantage over constraint store in multiple alldiff problems. Wider MDDs yield greater speedups (factor of 30). No backtracking, even with narrow MDDs. MDD store is slower than MILP for separable equalities, but much faster than using inequality propagator. Slide 53

54 Conclusions Intensive processing at search tree nodes can pay off when constraint store is richer. Ideally, use integrated problem solving Continuous relaxation, cutting planes, etc. MDD propagation, filtering, in parallel. Slide 54

55 Research Issues Are MDDs superior for optimization problems? Optimal solution = shortest path to terminus How to adjust width of relaxed MDD? Can always set width = 1 if MDDs not useful. How to propagate other global constraints in an MDD-based constraint store? Regular constraint (and special cases, pattern & stretch) Sequence constraint (and special case, among) Slide 55

Propagating separable equalities in an MDD store

Propagating separable equalities in an MDD store T. Hadzic 1, J. N. Hooker 2, and P. Tiedemann 3 1 University College Cork t.hadzic@4c.ucc.ie 2 Carnegie Mellon University john@hooker.tepper.cmu.edu 3 IT