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Introduction

Introduction. Combining two frameworks AND/OR Search Spaces Multi-valued Decision Diagrams ( MDDs ) Both are more compact ways to represent problems. Their combination yields an even more compact representation. Decision Diagrams are known to allow online speed queries. OFFLINE. A. F. B.

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Introduction

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  1. Introduction • Combining two frameworks • AND/OR Search Spaces • Multi-valued Decision Diagrams (MDDs) • Both are more compact ways to represent problems. • Their combination yields an even more compact representation. • Decision Diagrams are known to allow online speed queries.

  2. OFFLINE A F B C E D ONLINE Evaluate answer Polytime? Constant time? Introduction Query Compiled structure ONCE MANY

  3. A A A A A A A B B B B B B B B B B B B C C C C C C C C C C C C C C C C C 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 Decision tree OBDD [Bryant86] Ordered Binary Decision Diagram B = {0,1} f : B3→ B Table 1) Merge nodes with identical children Merge terminal nodes Simplify 2) Remove redundant nodes Ordering enables efficient operations

  4. A B B A C C A C C C C B D D B B B D D D D D D D D D D G G E E E G G E E E E E E E E E H F H F F F F F 0 0 0 0 1 1 1 1 G G H 0 1 AND Decision Diagrams f = (A ∨ E) ∧ (B ∨ F) ? f ∧ g = OBDD g = (C ∨G) ∧ (D ∨ H) f ∧ g = OBDD

  5. A 0 1 A B B 0 1 0 1 0 1 B B C 1 0 1 0 1 C 0 1 1 Minimal AND/OR graph Decision Diagram AOMDDs A Point dead-ends to terminal node “0” Point goods to terminal node “1” B C

  6. A A redundant A A 0 1 0 1 0 0 1 1 B B B B B B 0 1 0 1 0 1 0 0 1 1 0 1 C C C C 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 OBDD (pseudo tree is a chain) Removing Redundancy A Group OR node together with its AND children into a meta-node B C

  7. A D B B D D D A B 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 C C 0 0 1 1 0 1 0 1 0 1 AOBDD A A A B B B C D D C = *

  8. B D D A B A 0 0 0 0 0 0 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 A C B 0 1 D C Apply Operator A1B1 A1 B1 A1B1 = * A2B2 A4B6 A2B2 A2 B2 B6 A4B4 A4 B4 B7 0 1 0 1 A3B3 0 A3 A5 B3 B5

  9. B D D A B A 0 0 0 0 0 0 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 A C B 0 1 D C Apply Operator A1 B1 A1B1 = * A4B6 A2B2 A2 B2 B6 A4 A4B4 B4 A4 B4 B7 0 1 0 1 0 A3 A5 B3 B5

  10. A B D D B A D 0 0 0 0 0 0 0 1 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 A C C B 0 0 1 1 D C Apply Operator A1 B1 A1B1 = * A4B6 A2B2 A2 B2 B6 A4 B4 A4 B4 A4 B4 B7 0 1 0 1 0 1 A3 A5 B3 B5

  11. D B D D B A B A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 A C C B 0 0 1 1 D C Apply Operator A1 B1 A1B1 = * A4B6 A2B2 A4B6 A2 B2 B6 A4B4 A4B7 A4 B4 A4 B4 B7 0 1 0 1 0 1 A3 A5 B3 B5

  12. B B D D D B A D A 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 A 0 1 B B 0 0 1 1 A C C B 0 0 1 1 D C Apply Operator A1 B1 A1B1 = * A2B2 A4B6 A2 B2 B6 A4B7 A4 B4 A4 B4 B7 B7 0 1 0 1 0 1 A3 A5 B3 B5

  13. And/Or Multi-Valued Decision Diagrams • AOMDDs are: • AND/OR search graphs • canonical representations, given a pseudo tree • Defined by two rules: • All isomorphic subgraphs are merged • There are no redundant (meta) nodes

  14. A A D G B C B F C F A m7 E A H D E G H B primal graph pseudo tree B C F 0 1 m6 m6 m3 0 1 m3 D E G H A A B F 0 1 0 1 B C F C9 C5 0 1 0 1 0 1 m2 C m4 m1 m5 0 1 m1 m5 m4 G H m2 D E A A A C C B A B C C D D C B C F H E A C B C F F E C H B B F A H D G G G A D B G F H F B A D D A B C E F E H A A F B G H C G F B G F 0 1 0 1 0 1 0 1 0 1 0 1 F E H D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C8 E G D C1 C4 C6 C7 C2 F C3 0 1 H G (f ∨ h) ∧ (a∨!h) ∧ (a#b#g) ∧ (f ∨ g) ∧ (b ∨ f) ∧ (a ∨ e) ∧ (c ∨ e) ∧ (c#d) ∧ (b ∨ c) Example: m7

  15. A m7 B C F 0 1 m6 A 0 1 m3 D E G H 0 1 m7 B G B C C C H A F C B F F G H B A F B D C F C C D E F B A H B D F F E D C H G G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Example (continued)

  16. D G C B F A E A H B primal graph C F D E G H 0 1 C G F F C B G F H A B D C E H F D C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 AOBDD vs. OBDD A B B C C C C D D D D D D E E E F F F F G G G G G A H H B C 1 0 D AOBDD 18 nonterminals 47 arcs OBDD 27 nonterminals 54 arcs E F G H

  17. [ ] A A f1(ABC) [A] B f2(ABD) B D f3(BDE) [AB] [AB] C D [BD] C E E A 0 1 B B 0 1 0 1 C D C D C D C D 8 8 2 2 2 5 2 5 8 1 8 0 8 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E E E E E 3 4 3 4 3 4 3 4 8 8 8 8 8 8 8 8 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Constraint Optimization - AND/OR Tree

  18. [ ] A A f1(ABC) [A] B f2(ABD) B D f3(BDE) [AB] [AB] C D [BD] C E E A 0 1 B B 0 1 0 1 C D C D C D C D 8 8 2 2 2 5 2 5 8 1 8 0 8 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E 3 4 3 4 8 8 8 8 0 1 0 1 0 1 0 1 AND/OR Context Minimal Graph

  19. A 0 1 B 0 1 C C D 8 2 2 8 8 0 0 1 0 1 0 1 E E 3 4 8 8 0 1 0 1 AOMDD – Compilation by Search A A 0 1 B B B 0 1 0 1 C D C D C D C D C D 2 2 2 5 2 5 8 8 8 8 1 8 0 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E E 3 4 3 4 8 8 8 8 0 1 0 1 0 1 0 1 redundant 0 1

  20. A 0 1 B B 0 1 0 1 C D C D 2 2 2 5 8 0 8 6 0 1 0 1 0 1 0 1 E E E E 3 3 4 4 8 8 8 8 0 0 1 1 0 0 1 1 AOMDD – Compilation by Search A A 0 1 B B B 0 1 0 1 C D C D C D C D C D 2 2 2 5 2 5 8 8 8 8 1 8 0 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E E 3 4 3 4 8 8 8 8 0 1 0 1 0 1 0 1 isomorphic isomorphic isomorphic 0 1

  21. A 0 1 B B 0 1 0 1 C D D C D 2 2 5 2 5 8 0 6 8 6 0 1 0 1 0 1 0 1 0 1 E E 3 4 8 8 0 1 0 1 AOMDD – Compilation by Search A A 0 1 B B B 0 1 0 1 C D C D C D C D C D 2 2 2 5 2 5 8 8 8 8 1 8 0 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E E 3 4 3 4 8 8 8 8 0 1 0 1 0 1 0 1 isomorphic isomorphic 0 1

  22. A 0 1 B B 0 1 0 1 C D 2 2 8 0 0 1 0 1 E E 3 4 8 8 0 1 0 1 D 5 6 0 1 AOMDD – Compilation by Search A A 0 1 B B B 0 1 0 1 C D C D C D C D C D 2 2 2 5 2 5 8 8 8 8 1 8 0 6 8 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E E E E 3 4 3 4 8 8 8 8 0 1 0 1 0 1 0 1 redundant 0 1

  23. A A 0 0 1 1 B B 0 0 1 1 C C D D 2 2 2 2 8 8 0 0 0 0 1 1 0 0 1 1 E E E E 3 3 4 4 8 8 8 8 0 0 1 1 0 0 1 1 0 1 0 1 AOMDD for two best solutions D A=0 A=0 A=1 A=1 A=1 A=1 5 6 0 1 B=1 B=1 B=0 B=1 B=0 B=1 2 2 2 2 6 2 6 2 5 2 5 C=1 D=1 C=1 D=0 C=1 D=0 C=1 D=0 C=1 D=1 C=1 D=1 3 4 3 3 4 4 E=1 E=1 E=1 E=1 E=1 E=1 AOMDD for Constraint Optimization AOMDD for all solutions cost = 11 cost = 5 cost = 8 cost = 11 cost = 11 cost = 11

  24. Complexity of Compilation • The size of the AOMDD is O(n kw*) • The compilation time is also bounded by O(n kw*) • k = domain size • n = number of variables • w*= treewidth

  25. Semantic Treewidth • Given a network, there may exist a sparser equivalent network. • Challenges the idea of using induced width to measure the difficulty of the problem • AOMDD sizes are much smaller than the bound

  26. Semantic Treewidth • With respect to a pseudo tree, this is the smallest treewidth over all equivalent networks that can have that pseudo tree • With respect to the network, this is the smallest semantic treewidth over all pseudo trees that can express the set of solutions • Instead of the induced width bounding AOMDD size, we can use semantic treewidth.

  27. Semantic Treewidth Equality constraint network results

  28. Constraint Propagation • We can also prune the search space during compilation without removing possible solutions. • In Bayesian networks, prune a subtree if the weight of the assignment is 0.

  29. Experiments • What about the pseudo-tree height parameter? • Problems: WCSP instances • ISCAS 89 Circuits • SPOT5 Satellites • Mastermind • CELAR6 Radio Frequencies • Time bound for compilation: 3 hours

  30. Experiments • Compilation was for finding the optimal solution • Used AOBB with static mini-bucket heuristics (i-bound = 10) • Tried different implementations of MinFill • Existing implementation in the compiler • daoopt(gets lower h because it considers it too) • CVO

  31. Experiments • BnBpruning makes the size unpredictable as a function of the parameters • Need to modify the routine for solution counting so the entire AOMDD is actually compiled

  32. Experiments (BN) • Reproduce and extend BN results in JAIR 2008 paper • UAI 2006 Bayesian network benchmarks • Domain sizes of 2 • Evidence on 30 random variables (to simplify the networks slightly) • Many elements with “0” support • Compile with constraint propagation

  33. To do • Perform experiments to compare optimization vs. full compilation • Need to extend code for WCSPs • Another way to deal with unpredictability of w/h vs. size in optimization? • Compute many orderings with equal w/h and average the search space/AOMDD sizes.

  34. Future Work? • Try regressing curves that depend on w, h, or both • Evaluation of bottom-up version of compilation (Robert’s algorithm in CP 2006)

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