Constraint-based problem solving

1. Constraint-based problem solving • Model problem • specify in terms of constraints on acceptable solutions • define variables (denotations) and domains • define constraints in some language • Solve model • define search space / choose algorithm • incremental assignment / backtracking search • complete assignments / stochastic search • design/choose heuristics • Verify and analyze solution

2. Constraint-based problem solving • Model problem • specify in terms of constraints on acceptable solutions • define variables (denotations) and domains • define constraints in some language • Solve model • define search space / choose algorithm • incremental assignment / backtracking search • complete assignments / stochastic search • design/choose heuristics • Verify and analyze solution Constraint Satisfaction Problem

3. Constraint satisfaction problem • A CSP is defined by • a set of variables • a domain of values for each variable • a set of constraints between variables • A solution is • an assignment of a value to each variable that satisfies the constraints

4. Example: Crossword puzzles

5. Options CSP, binary CSP, SAT, 3-SAT, ILP, ... • Model and solve in one of these languages • Model in one language, translate into another to solve

6. Article of faith “Constraints arise naturally in most areas of human endeavor. They are the natural medium of expression for formalizing regularities that underlie the computational and physical worlds and their mathematical abstractions.” P. Van Hentenryck and V. Saraswat Constraint Programming: Strategic Directions

7. CSP 3-SAT binary CSP ILP (0,1)-ILP Reducibility NP-Complete

8. Options CSP, binary CSP, SAT, 3-SAT, ILP, ... • Model and solve in one of these languages • Model in one language, translate into another to solve

9. Importance of the model “In integer programming, formulating a ‘good’ model is of crucial importance to solving the model.” G. L. Nemhauser and L. A. Wolsey Handbook in OR & MS, 1989 “Same for constraint programming.” Folk Wisdom, CP practitioners

10. Measures for comparing models • How easy is it to • write down, • understand, • modify, debug, • communicate? • How computationally difficult is it to solve?

11. Constraint systems/languages

12. Abstractions • Capture commonly occurring constraints • special propagation algorithms • e.g. alldifferent, cardinality, cumulative, ... • User-defined constraints • full power of host language

13. CSP versus ILP • OO databases versus Relational databases • C++ versus C • C++ versus Fortran • C++ versus Assembly language

14. Computational difficulty? • What is a good model depends on algorithm • Choice of variables defines search space • Choice of constraints defines • how search space can be reduced • how search can be guided

15. Improving model efficiency Given a model: • Add/remove variables, values, constraints (keep the denotation of the variables) • Use/translate to a different representation (change the denotation of the variables)

16. Improve computational efficiency of model by adding “right” constraints symmetries removed dead-ends encountered earlier in search process Three methods: add hand-crafted constraints during modeling apply a consistency algorithm before solving learn constraints while solving Adding redundant constraints

17. Adding hand-crafted constraints: crossword puzzles Applying a consistency algorithm: dual representation Examples:

18. Variables that are abstractions of other variables e.g, decision variables Suppose x has domain {1,…,10} Add Boolean variable to represent decisions (x < 5), (x 5) Variables that represent constraints Adding redundant variables

19. Translations between models • Improve computational efficiency by completely changing the model • change denotation of variables • e.g, convert from non-binary to binary • aggregate variables • e.g, timetabling multiple sections of a course • Not much as has been done on either the theory or the practice side

20. Conversion to binary • Any CSP can be converted into one with only binary constraints • Two techniques known • dual graph method (Dechter & Pearl, 1989) • hidden variable method (Peirce, 1933; Dechter 1990) • Translations are polynomial if constraints are represented extensionally

21. 3-SAT example (x1 x2  x6)  (x1  x3  x4)  (x4  x5  x6)  (x2  x5 x6) • Non-binary CSP: • Boolean variables: x1, …, x6 • constraints: one for each clause C1(x1, x2, x6) = {(0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)} C2(x1, x3, x4) = ...

22. Dual graph representation x2, x6 y1 = (x1, x2, x6) y4 = (x2, x5, x6) x6 x1 x5, x6 y2 = (x1, x3, x4) y3 = (x4, x5, x6) x4

23. Hidden variable representation h1 h4 x1 x2 x3 x4 x5 x6 h3 h2

24. Hidden variable representation • Non-binary constraint: • Augmented constraint: • Binary constraints: C1(x1, x2, x6) = {(0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)} C1(x1, x2, x6,h1) = {(0,0,1,0), (0,1,0,1), (0,1,1,2), (1,0,0,3), (1,0,1,4), (1,1,0,5), (1,1,1,6)} R1(x1, h1) = {(0,0), (0,1), (0,2), (1,3), (1,4), (1,5), (1,6)} R2(x2, h1) = {(0,0), (1,1), (1,2), (0,3), (0,4), (1,5), (1,6)} R3(x6,h1) = {(1,0), (0,1), (1,2), (0,3), (1,4), (0,5), (1,6)}

25. FC on non-binary: O(n) consistency checks FC on dual: O(2n) consistency checks Dual exponentially worse x1, x1  x2, x1  x2  x3, ..., x1  …  xn-1  xn x1 h1 x2 h2 x3 h3

26. FC on non-binary: O(n2n) consistency checks FC on dual: O(n2) consistency checks Dual exponentially better x1  …  xn-1, x1  …  xn-2  xn, …, x2  …  xn x1 h1 x2 h2 x3 h3

27. Bounds on performance • Worst case: • Best case: FC(non-binary)  FC+(hidden) dk FC+(hidden)  FC(non-binary) dn

28. Crossword puzzles 1 2 3 4 5 a aardvark aback abacus abaft abalone abandon ... Mona Lisa monarch monarchy monarda ... zymurgy zyrian zythum 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

29. Crossword puzzles

30. Crossword puzzles

31. Model of random non-binary CSP • n variables • each with domain size d • m constraints • each with k variables, chosen at random • each with t tuples, chosen at random

32. Order of magnitude curves n = 20 d = 2 k = 3 n = 20 d = 10 k = 3 (number of constraints) / (number of variables) n = 20 d = 2 k = 5 n = 20 d = 10 k = 5 (number of tuples in constraints) / (maximum tuples)

33. Random 3-SAT

34. Project: Planning • Given start state, goal state, and actions: determine a plan (a sequence of actions) Box1 Box2 Box2

35. Contrasts • CP philosophy/methodology • emphasis on modeling, domain knowledge • general purpose search algorithm • backtracking with constraint propagation • Successful • e.g., can solve practical scheduling problems

36. Contrasts • Planning philosophy/methodology • emphasis on minimal model • just representation of actions • special-purpose search algorithms • Not as successful • has not solved many practical problems

37. Tradeoffs • Robust CSP model needed for each new domain • can require much intellectual effort • Less work needs to be done on algorithms • many general purpose constraint solvers available

38. CSP model of planning • State-based model • model each state by a collection of variables • constraints enforce valid transitions between states • Example: logistics world • variable for each package, truck, plane • domains of packages: all locations, trucks, planes • domains of trucks, planes: all locations

39. CSP model: constraints (I) • Action constraints • model the effects of actions • patterned after explanation closure axioms • State constraints • variables within a state must be consistent

40. Improving model efficiency • Can add/remove/aggregate/decompose • variables • domain values • constraints • Here: • added hidden variables • added redundant & symmetry-breaking constraints

41. CSP model: constraints (II) • Symmetric values constraints • break symmetries on values variables can be assigned • Action choice constraints • break symmetries on equivalent permutations of actions

42. CSP model: constraints (III) • Domain constraints • restrictions on original domains of variables • Capacity constraints • bounds on resources • Distance constraints • bounds on steps needed for a variable to change from one value to another

43. Solving the CSP model • Given an instance of a planning problem • generate a model with one step in it • instantiate variables in the initial and goal states • search for a solution (GAC+CBJ) • repeat, incrementing number of steps, until plan is found • Properties: • forwards, backwards, or middle out planner • sound, complete, guaranteed to terminate

44. Experiments • Five test domains from AIPS’98 • Five planners: • CPlan • Blackbox (Kautz & Selman) • HSP (Bonet & Geffner) • IPP (Koehler & Nebel) • TLPlan (Bacchus & Kabanza) • Setup: • machines: 400MHz Pentium II’s • resources: 1 hour CPU time, 256 Mb memory

45. Gripper problems

46. Logistics problems

47. Mystery problems

48. MysteryPrime problems

49. Grid problems

50. Related work • Planning as a CSP • satisfiability (Kautz & Selman) • ILP (e.g., Bockmayr & Dimopoulos) • Adding declarative knowledge to improve efficiency • hand-coded (e.g, Kautz & Selman, Bacchus & Kabanza) • automatically derived (e.g., Fox & Long, Nebel et al.)