Introduction & Overview

Transcription

1 ID2204: Constraint Programming Introduction & Overview Lecture 01, Christian Schulte Software and Computer Systems School of Information and Communication Technology KTH Royal Institute of Technology Sweden

2 Lecture Overview What is Constraint Programming? Sudoku is Constraint Programming... more later

3 Sudoku...is Constraint Programming!

4 Sudoku Assign blank fields digits such that: digits distinct per rows, columns, blocks

5 Sudoku Assign blank fields digits such that: digits distinct per rows, columns, blocks

6 Sudoku Assign blank fields digits such that: digits distinct per rows, columns, blocks

7 Sudoku Assign blank fields digits such that: digits distinct per rows, columns, blocks

8 Block Propagation No field in block can take digits 3,6,8

9 Block Propagation 1,2,4,5,7,9 8 1,2,4,5,7,9 1,2,4,5,7, ,2,4,5,7,9 1,2,4,5,7,9 1,2,4,5,7,9 No field in block can take digits 3,6,8 propagate to other fields in block Rows and columns: likewise

10 Propagation ,2,3,4,5,6,7,8, Prune digits from fields such that: digits distinct per rows, columns, blocks

11 Propagation ,3,5,6,7, Prune digits from fields such that: digits distinct per rows, columns, blocks

12 Propagation ,3,6,7 Prune digits from fields such that: digits distinct per rows, columns, blocks

13 Propagation ,3, Prune digits from fields such that: digits distinct per rows, columns, blocks

14 Iterated Propagation Iterate propagation for rows, columns, blocks What if no assignment: search... later

15 Sudoku is Constraint Programming Variables: fields take values: digits maintain set of possible values Constraints: distinct relation among variables Modelling: variables, values, constraints Solving: propagation, search

16 Constraint Programming Variable domains finite domain integer, finite sets, multisets, intervals,... Constraints distinct, arithmetic, scheduling, graphs,... Solving propagation, branching, exploration,... Modelling variables, values, constraints, heuristics, symmetries,...

17 Plan of Lecture Introduction what is constraint programming? principles and applications Overview course content course goal Organizational

18 What Is Constraint Programming?

19 Running Example: SMM Find distinct digits for letters, such that SEND + MORE = MONEY

20 Constraint Model for SMM Variables: S,E,N,D,M,O,R,Y {0,,9} Constraints: distinct(s,e,n,d,m,o,r,y) 1000 S+100 E+10 N+D M+100 O+10 R+E = M+1000 O+100 N+10 E+Y S 0 M 0

21 Solving SMM Find values for variables such that all constraints satisfied

22 Finding a Solution Compute with possible values rather than enumerating assignments Prune inconsistent values constraint propagation Search branch: define search tree explore: explore search tree for solution

23 Constraint Propagation

24 Important Concepts Constraint store Propagator Constraint propagation

25 Constraint Store x {3,4,5} y {3,4,5} Maps variables to possible values stores basic constraints

26 Constraint Store finite domain constraints x {3,4,5} y {3,4,5} Maps variables to possible values Others: finite sets, intervals, trees,...

27 Propagators Implement (non-basic) constraints distinct(x 1,,x n ) x + 2*y = z

28 Propagators x y y>3 x {3,4,5} y {3,4,5} Amplify store by constraint propagation

29 Propagators x y y>3 x {3,4,5} y {3,4,5} Amplify store by constraint propagation

30 Propagators x y y>3 x {3,4,5} y {4,5} Amplify store by constraint propagation

31 Propagators x y y>3 x {3,4,5} y {4,5} Amplify store by constraint propagation

32 Propagators x y y>3 x {4,5} y {4,5} Amplify store by constraint propagation

33 Propagators x y y>3 x {4,5} y {4,5} Amplify store by constraint propagation Disappear when done (subsumed, entailed) no more propagation possible

34 Propagators x y x {4,5} y {4,5} Amplify store by constraint propagation Disappear when done (subsumed, entailed) no more propagation possible

35 Propagation for SMM Results in store S {9} E {4,,7} M {1} O {0} N {5,,8} D {2,,8} R {2,,8} Y {2,,8} Propagation alone not sufficient! create simpler sub-problems branching

36 Constraints and Propagators Constraints state relations among variables which value combinations satisfy constraint Propagators implement constraints prune values in conflict with constraint Constraint propagation drives propagators for several constraints

37 Search

38 Important Concepts Branching Exploration Branching heuristics Best-solution search

39 Search: Branching x y x {4,5} y {4,5} x=4 x 4 x y x {4} y {4} x y x {5} y {4,5} Create subproblems with additional information enable further constraint propagation

40 Example Branching Strategy Pick variable x with at least two values Pick value n from domain of x Branch with x=n and x n Part of model

41 Search: Exploration Iterate propagation and branching Orthogonal: branching exploration Nodes: Unsolved Failed Succeeded

42 SMM: Unique Solution SEND + MORE = MONEY = 10652

43 Heuristics for Branching Which variable least possible values (first-fail) application dependent heuristic Which value minimum, median, maximum x=m or x m split with median m x<m or x m Problem specific

44 SMM: Solution With First-fail SEND + MORE = MONEY = 10652

45 Send Most Money (SMM++) Find distinct digits for letters, such that SEND + MOST = MONEY and MONEY maximal

46 Best Solution Search Naïve approach: compute all solutions choose best Branch-and-bound approach: compute first solution add betterness constraint to open nodes next solution will be better prunes search space

47 Branch-and-bound Search Find first solution

48 Branch-and-bound Search Explore with additional constraint

49 Branch-and-bound Search Explore with additional constraint

50 Branch-and-bound Search Guarantees better solutions

51 Branch-and-bound Search Guarantees better solutions

52 Branch-and-bound Search Last solution best

53 Branch-and-bound Search Proof of optimality

54 Modelling SMM++ Constraints and branching as before Order among solutions with constraints so-far-best solution S,E,N,D,M,O,T,Y current node S,E,N,D,M,O,T,Y constraint added M+1000 O+100 N+10 E+Y < M+1000 O+100 N+10 E+Y

55 SMM++: Branch-and-bound SEND + MOST = MONEY = 10876

56 SMM++: All Solution Search SEND + MOST = MONEY = 10876

57 Summary

58 Summary: Key Ideas and Principles Modelling variables with domain constraints to state relations branching strategy solution ordering Solving constraint propagation constraint branching search tree exploration applications principles

59 Widely Applicable Timetabling Scheduling Crew rostering Resource allocation Workflow planning and optimization Gate allocation at airports Sports-event scheduling Railroad: track allocation, train allocation, schedules Automatic composition of music Genome sequencing Frequency allocation

60 Draws on Variety of Techniques Artificial intelligence basic idea, search,... Operations research scheduling, flow,... Algorithms graphs, matching, networks,... Programming languages programmability, extensionability,...

61 Essential Aspect Compositional middleware for combining smart algorithmic problem substructures components (propagators) scheduling graphs flows plus essential extra constraints

62 Principles Models for constraint propagation properties and guarantees Strong constraint propagation global constraints with strong algorithmic methods mantra: search kills, search kills, search k Branching strategies Exploration strategies

63 SMM: Strong Propagation SEND + MORE = MONEY = 10652

64 Scheduling Resources Modelling Propagation Strong propagation

65 Scheduling Resources: Problem Tasks duration resource Precedence constraints determine order among two tasks Resource constraints at most one task per resource [disjunctive, non-preemptive scheduling]

66 Scheduling: Bridge Example Infamous: additional side constraints!

67 Scheduling: Solution Start time for each task All constraints satisfied Earliest completion time minimal make-span

68 Scheduling: Model Variable for start-time of task a start(a) Precedence constraint: a before b start(a) + dur(a) start(b)

69 Propagating Precedence a before b a b start(a) {0,,7} start(b) {0,,5}

70 Propagating Precedence a before b a a b b start(a) {0,,7} start(b) {0,,5} start(a) {0,,2} start(b) {3,,5}

71 Scheduling: Model Variable for start-time of task a start(a) Precedence constraint: a before b start(a) + dur(a) start(b) Resource constraint: a before b or b before a

72 Scheduling: Model Variable for start-time of task a start(a) Precedence constraint: a before b start(a) + dur(a) start(b) Resource constraint: start(a) + dur(a) start(b) or b before a

73 Scheduling: Model Variable for start-time of task a start(a) Precedence constraint: a before b start(a) + dur(a) start(b) Resource constraint: start(a) + dur(a) start(b) or start(b) + dur(b) start(a)

74 Reified Constraints Use control variable b {0,1} c b=1 Propagate c holds propagate b=1 c holds propagate b=0 b=1 holds propagate c b=0 holds propagate c

75 Reified Constraints Use control variable b {0,1} c b=1 Propagate not easy! c holds propagate b=1 c holds propagate b=0 b=1 holds propagate c b=0 holds propagate c

76 Reification for Disjunction Reify each precedence [start(a) + dur(a) start(b)] b 0 =1 and [start(b) + dur(b) start(a)] b 1 =1 Model disjunction b 0 + b 1 1

77 Model Is Too Naive Local view individual task pairs O(n 2 ) propagators for n tasks Global view ("global" constraints) all tasks on resource single propagator smarter algorithms possible

78 Example: Edge Finding Find ordering among tasks ( edges ) For each subset of tasks {a} B assume: a before B deduce information for assume: B before a deduce information for join computed information can be done in O(n 2 ) a and B a and B

79 Summary Modeling easy but not always efficient constraint combinators (reification) global constraints smart heuristics More on constraint-based scheduling Baptiste, Le Pape, Nuijten. Constraint-based Scheduling, Kluwer, 2001.

80 Course Overview

81 Content Overview As to be expected, no surprises: applications principles pragmatics limitations

82 Modeling with CP Basic solving methods constraint propagation search Typical techniques for modeling in different application areas redundant constraints, symmetry elimination Refining models by strong algorithmic methods Heuristic search methods Application to hard real-size problems

83 Principles Underlying CP Models for propagation search and their essential properties Different levels of consistency (propagation strength) Different constraint domains finite domains, finite sets,...

84 Strong Algorithmic Methods Régin's distinct algorithm Edge-finding Integration achieving required properties for propagation

85 Relation to Other Techniques Integer programming Local search Discussion of merits and weaknesses Hybrid approaches

86 Goals: Learning Outcomes explain and apply basic modeling techniques for constraint problems, including the selection of variables, constraints, and optimization criteria. describe and apply depth-first search and branch-and-bound search for solving constraint problems. describe and define constraint propagation, search branching, and search tree exploration prove correctness, consistency, and completeness of propagators implementing constraints. define and prove correctness of branching strategies. describe optimizations of constraint propagation based on fixpoint reasoning.

87 Goals: Learning Outcomes describe advanced modeling techniques, analyze combinatorial problems for the applicability of these techniques, and apply and combine them techniques include: general symmetries, value and variable symmetries, symmetry breaking with constraints, symmetry breaking during search, domination constraints, redundant constraints, redundant modeling and channeling, using strong algorithmic techniques, and branching heuristics. describe and apply Régin's algorithm for the distinct constraint as an example of strong constraint propagation explain algorithms for the element constraint, linear constraints, and disjunctive scheduling constraints. implement a simple propagation algorithm. describe the main strength and weaknesses of constraint programming and how constraint programming relates to other methods (local search and integer programming)

88 Organizational

89 Material Lecture notes (slides) available before the lectures Additional material book excerpts scientific articles notes written by me Modeling and Programming with Gecode

90 How to Pass? Pass exam has 200 (3 hour exam) exam points 100 total points needed grading scale linear (see www) Total pts = exam pts + bonus pts Bonus points from assignments at most 20 points

91 Assignments Four assignments each 5 bonus points if submitted in time one to three weeks for solving Points only valid in this academic year!

92 Assignment Tasks Exploration tasks small tryouts need to be done in order to do Submission tasks submit in time, get bonus points do them, do them submit to me by Both practical and principles

93 Software: Gecode Gecode C++ library course web: links to Gecode page Download and install at least version (available on Monday) Start reading Modeling and Programming with Gecode at the beginning of Part M are reading instructions

94 Contacting Christian Schulte other options (urgent cases), see my homepage

95 Summary Constraint programming is exciting! is fun! Understanding of principles and applications necessary Read the webpage

96 Acknowledgments I am grateful to Pierre Flener for helpful comments and bugreports on these slides