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Foundations of Constraint Processing

Learn about the foundations of constraint processing, including the definition, representation, and solving techniques for constraint satisfaction problems. Explore practical applications and the power of constraint processing.

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Foundations of Constraint Processing

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  1. Constraint Satisfaction 101 Foundations of Constraint Processing CSCE421/821, Fall 2004: www.cse.unl.edu/~choueiry/F04-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B choueiry@cse.unl.edu Tel: +1(402)472-5444 Overview: 1

  2. Outline • Motivating example, application areas • CSP: Definition, representation • Some simple modeling examples • More on definition and formal characterization • Basic solving techniques • (Implementing backtrack search) • Advanced solving techniques • Issues & research directions Overview: 1

  3. Motivating example I • Context: You are a senior in college • Problem: You need to register in 4 courses for the Spring semester • Possibilities: Many courses offered in Math, CSE, EE, CBA, etc. • Constraints: restrict the choices you can make • Unary: Courses have prerequisites you have/don't have Courses/instructors you like/dislike • Binary: Courses are scheduled at the same time • n-ary: In CE: 4 courses from 5 tracks such as at least 3 tracks are covered • You have choices, but are restricted by constraints • Make the right decisions!! Overview: 1

  4. Motivating example II • Given • A set of variables: 4 courses at UNL • For each variable, a set of choices (values) • A set of constraints that restrict the combinations of values the variables can take at the same time • Questions • Does a solution exist? (classical decision problem) • How two or more solutions differ? How to change specific choices without perturbing the solution? • If there is no solution, what are the sources of conflicts? Which constraints should be retracted? • etc. Overview: 1

  5. Practical applications Adapted from E.C. Freuder • Radio resource management (RRM) • Databases (computing joins, view updates) • Temporal and spatial reasoning • Planning, scheduling, resource allocation • Design and configuration • Graphics, visualization, interfaces • Hardware verification and software engineering • HC Interaction and decision support • Molecular biology • Robotics, machine vision and computational linguistics • Transportation • Qualitative and diagnostic reasoning Overview: 1

  6. Constraint Processing • is about ... • Solving a decision problem… • … While allowing the user to state arbitrary constraints in an expressive way and • Providing concise and high-level feedback about alternatives and conflicts • Related areas: • AI, OR, Algorithmic, DB, TCS, Prog. Languages, etc. Overview: 1

  7. relax reinforce Power of Constraints Processing • Flexibility & expressiveness of representations • Interactivity users can constraints Overview: 1

  8. Outline • Motivating example, application areas • CSP: Definition, representation • Some simple modeling examples • More on definition and formal characterization • Basic solving techniques Overview: 1

  9. Definition of a Problem • General template of any computational problem • Given: • Example: a set of objects, their relations, etc. • Query/Question: • Example: Find x such that the condition y is satisfied • How about the Constraint Satisfaction Problem? Overview: 1

  10. Definition of a CSP • GivenP = (V, D, C ) • V is a set of variables, • D is a set of variable domains (domain values) • C is a set of constraints, • Query: can we find a value for each variable such that all constraints are satisfied? Overview: 1

  11. Other queries… • Find a solution • Find all solutions • Find the number of solutions • Find a set of constraints that can be removed so that a solution exists • Etc. Overview: 1

  12. Representation I • Given P = (V, D, C ), where • Find a consistent assignment for variables • Constraint Graph • Variable   node (vertex) • Domain  node label • Constraint  arc (edge) between nodes Overview: 1

  13. v1 < v2 V1 {1, 2, 3, 4} { 3, 6, 7 } V2 v1+v3 < 9 v2 > v4 v2 < v3 { 3, 4, 9 } { 3, 5, 7 } V3 V4 Representation II • Given P = (V, D, C ), where • Example I: • Define C ? Overview: 1

  14. Outline • Motivating example, application areas • CSP: Definition, representation • Some simple modeling examples • More on definition and formal characterization • Basic solving techniques Overview: 1

  15. Example II: Temporal reasoning • Give one solution: ……. • Satisfaction, yes/no: decision problem Overview: 1

  16. Example III: Map coloring Using 3 colors (R, G, & B), color the US map such that no two adjacent states have the same color • Variables? • Domains? • Constraints? Overview: 1

  17. Example III: Map coloring Using 3 colors (R, G, & B), color the US map such that no two adjacent states have the same color Overview: 1

  18. { a, b, c } { a, b } { a, c, d } { b, c, d } Example IV: Resource Allocation What is the CSP formulation? Overview: 1

  19. { a, b, c } { a, b } { a, c, d } { b, c, d } Constraint Graph T1 { R1, R3 } T4   T6 { R1, R2, R3 }   { R2, R4 }       { R1, R3 }  { R2, R4 }   { R1, R2, R3 } T7 T3 { R1, R3 } T5 Example IV: Resource Allocation Interval Order { R1, R3 } T1 { R1, R3 } T2 { R1, R3, R4 } { R1, R3 } T2 T4 T3 { R1, R2, R3 } T5 { R2, R4 } T6 { R2, R4 } T7 Overview: 1

  20. Example V: Puzzle • Given: • Four musicians: Fred, Ike, Mike, and Sal, play bass, drums, guitar and keyboard, not necessarily in that order. • They have 4 successful songs, ‘Blue Sky,’ ‘Happy Song,’ ‘Gentle Rhythm,’ and ‘Nice Melody.’ • Ike and Mike are, in one order or the other, the composer of ‘Nice Melody’ and the keyboardist. • etc ... • Query: Who plays which instrument and who composed which song? Overview: 1

  21. Example V: Puzzle • Formulation 1: • Variables: Bass, Drums, Guitar, Keyboard, Blue Sky, Happy Song • Gentle Rhythm and Nice Melody. • Domains: Fred, Ike, Mike, Sal • Constraints: … • Formulation 2: • Variables: Fred's-instrument, Ike's-instrument, …, • Fred's-song, Ikes's-song, Mike’s-song, …, etc. • Domains: • { bass, drums, guitar, keyboard } • { Blue Sky, Happy Song, Gentle Rhythm, Nice Melody} • Constraints: … Overview: 1

  22. Example VI: Product Configuration • Train, elevator, car, etc. • Given: • Components and their attributes (variables) • Domain covered by each characteristic (values) • Relations among the components (constraints) • A set of required functionalities (more constraints) • Find: a product configuration • i.e., an acceptable combination of components • that realizes the required functionalities Overview: 1

  23. v1 < v2 Constraint Graph V1 {1, 2, 3, 4} { 3, 6, 7 } T1 V2 { R1, R3 } T4   T6 { R1, R2, R3 } v1+v3 < 9   v2 > v4 v2 < v3 { R2, R4 }       { R1, R3 } { 3, 4, 9 } { 3, 5, 7 } V3 V4  { R2, R4 }   { R1, R2, R3 } T7 T3 { R1, R3 } T5 Examples of Constraint Types • Example I: algebraic constraints • Example II: (algebraic) constraints of bounded difference • Example III & IV: coloring, mutual exclusion, difference constraints • Example V & VI: elements of C must be made explicit Overview: 1

  24. More examples • Example VII:Databases • Join operation in relational DB is a CSP • View materialization is a CSP • Example VIII: Interactive systems • Data-flow constraints • Spreadsheets • Graphical layout systems and animation • Graphical user interfaces • Example IX: Molecular biology (bioinformatics) • Threading, etc Overview: 1

  25. Outline • Motivating example, application areas • CSP: Definition, representation • Some simple modeling examples • More on definition and formal characterization • Basic solving techniques Overview: 1

  26. V1 a, b   V3 V2 a, c b, c  Representation (again) Macrostructure G(P): - constraint graph for binary constraints - constraint network for non-binary constraints Micro-structure (P): Co-microstructure co-(P): (V1, a ) (V1, b) (V2, a ) (V2, c) (V3, b ) (V3, c) no goods (V1, a ) (V1, b) (V2, a ) (V2, c) (V3, b ) (V3, c) Overview: 1

  27. v1 < v2 V1 {1, 2, 3, 4} { 3, 6, 7 } V2 v1+v3 < 9 v2 > v4 v2 < v3 { 3, 4, 9 } { 3, 5, 7 } V3 V4 Constraint arity I • Given P = (V, D, C ) , where • How to represent the constraint V1 + V2 + V4 < 10? Overview: 1

  28. Constraint arity II • Given P = (V, D, C ) , where Constraints: universal, unary, binary, ternary, … , global. A Constraint Network: Overview: 1

  29. Domain types • P = (V, D, C ) where • Domains: • Finite (discrete), enumeration works • Continuous, sophisticated algebraic techniques are needed Overview: 1

  30. Constraint terminology • Arity: • universal, unary, binary, ternary, …, global • Scope: • The set of variables to which the constraint applies • Definition: • Intention, extension • Implementation: • predicate, set of tuples, binary matrix, etc. Overview: 1

  31. Complexity of CSP Characterization: • Decision problem • In general, NP-complete by reduction from 3SAT Overview: 1

  32. Proving NP-completeness • Show that 1 is in NP • Given a problem 1 in NP, show that an known NP-complete problem 2 can be efficiently reduced to 1 • Select a known NP-complete problem 2 (e.g., SAT) • Construct a transformationf from 2 to 1 • Prove that f is a polynomial transformation (Check Chapter 3 of Garey & Johnson) Overview: 1

  33. What is SAT? Given a sentence: • Sentence: conjunction of clauses • Clause: disjunction of literals • Literal: a term or its negation • Term: Boolean variable Question: Find an assignment of truth values to the Boolean variables such the sentence is satisfied. Overview: 1

  34. CSP is NP-Complete • Verifying that an assignment for all variables is a solution • Provided constraints can be checked in polynomial time • Reduction from 3SAT to CSP • Many such reductions exist in the literature (perhaps 7 of them) Overview: 1

  35. Problem reduction Example: CSP into SAT (proves nothing, just an exercise) Notation: variable-value pair = vvp • vvp  term • V1 = {a, b, c, d} yields x1 = (V1, a), x2 = (V1, b), x3 = (V1, c), x4 = (V1, d), • V2 = {a, b, c} yields x5 = (V2, a), x6 = (V2, b), x7 = (V2,c). • The vvp’s of a variable  disjunction of terms • V1 = {a, b, c, d} yields • (Optional) At most one VVP per variable Overview: 1

  36. CSP into SAT (cont.) Constraint: • Way 1: Each inconsistent tuple  one disjunctive clause • For example: how many? • Way 2: • Consistent tuple  conjunction of terms • Each constraint  disjunction of these conjunctions  transform into conjunctive normal form (CNF) Question: find a truth assignment of the Boolean variables such that the sentence is satisfied Overview: 1

  37. Outline • Motivating example, application areas • CSP: Definition, representation • Some simple modeling examples • More on definition and formal characterization • Basic solving techniques • Modeling and consistency checking • Constructive, systematic search • Iterative improvement, local search Overview: 1

  38. How to solve a CSP? Search • 1. Constructive, systematic • 2. Iterative repair, local search Overview: 1

  39. Before starting search! Consider: • Importance of modeling/formulation: • To control the size of the search space • Preprocessing • A.k.a. constraint filtering/propagation, consistency checking • reduces size of search space Overview: 1

  40. 1 2 3 4 V1 V2 V3 V4 4 Rows 4 Column positions V11 V12 V13 V14 16 Cells V21 V22 V23 V24 V31 V32 V33 V34 V41 V42 V43 V44 Importance of modeling • N-queen: formulation 1 • Variables? • Domains? • Size of CSP? • N-queens: formulation 2 • Variables? • Domains? • Size of CSP? {0,1} Overview: 1

  41. 13 1- B: [ 5 .. 14 ] 14 C: [ 6 .. 15 ] 2- A: [ 2 .. 10 ] 2 C: [ 6 .. 14 ] 14 6 Constraint checking • Arc-consistency B A < B A [ 5.... 18] B < C [ 1.... 10 ] 2 < C - A < 5 3- B: [ 5 .. 13 ] [ 4.... 15] C Overview: 1

  42. Constraint checking • Arc-consistency: every combination of two adjacent variables • 3-consistency, k-consistency (k n) • Constraint filtering, constraint checking, etc.. • Eliminate non-acceptable tuples prior to search • Warning: arc-consistency does not solve the problem still is not a solution! Overview: 1

  43. Systematic search • Starting from a root node • Consider all values for a variable V1 • For every value for V1, consider all values for V2 • etc.. • For n variables, each of domain size d • Maximum depth? fixed! • Maximum number of paths? size of search space, size of CSP Overview: 1

  44. Systematic search (I)Back-checking • Systematic search generates dnpossibilities • Are all possibilities acceptable? • Expand a partial solution only when it is consistent • This yields early pruning of inconsistent paths Overview: 1

  45. Systematic search (II)Chronological backtracking What if only one solution is needed? • Depth-first search & Chronological backtracking • DFS: Soundness? Completeness? Overview: 1

  46. Systematic search (III)Intelligent backtracking What if the reason for failure was higher up in the tree? Backtrack to source of conflict !! • Backjumping, conflict-directed backjumping, etc. Overview: 1

  47. Systematic search (IV) Ordering heuristics • Which variable to expand first? • Heuristics: • most constrained variable first (reduce branching factor) • most promising value first (find quickly first solution) Overview: 1

  48. 3 1 2 2 2 4 1 3 4 1 3 1 4 1 2 4 4 1 1 2 3 3 2 2 Q Q Q Q Q Q Q 1 3 Solution! 26 nodes visited. Systematic search(V)Back-checking • Search tree with only backtrack search? Root node Q Overview: 1

  49. 2 1 1 3 4 3 4 2 Q Q Q Q Q Domain Wipe Out V3 Q Q Domain Wipe Out V4 Q Solution! 8 nodes visited. Systematic search(VI)Forward checking Search Tree with domains filter by Forward Check Root node Overview: 1

  50. Summary of backtrack search • Constructive, systematic, exhaustive • In general sound and complete • Back-checking: expands nodes consistent with past • Backtracking: Chronological vs. intelligent • Ordering heuristics: • Static • Dynamic variable • Dynamic variable-value pairs • Looking ahead: • Partial look-ahead • Forward checking (FC) • Directional arc-consistency (DAC) • Full (a.k.a. Maintaining Arc-consistency or MAC) Overview: 1

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