Look Back Methods

1. Look Back Methods Constraint Propagation • intelligent backtracking • consistency checks among instantiated variables • backjumping • backtracks to the conflicting variable • backchecking and backmarking • avoids redundant constraint checkingby remembering conflicting levelfor each value jump here a conflict b b b still conflict

2. Look Ahead Methods Constraint Propagation • preventing future conflicts via consistency checks among not yet instantiated variables • forward checking (FC) • AC to direct neighbourhood • partial look ahead (PLA) • DAC • (full) look ahead (LA) • Arc Consistency • Path Consistency instantiated variable labelling order

3. Look Ahead - Example Constraint Propagation • Problem:X::{1,2}, Y::{1,2}, Z::{1,2}X = Y, X  Z, Y > Z generate & test - 7 stepsbacktracking - 5 stepspropagation - 2 steps

4. Stochastic and Heuristic Methods • GT + smart generator of complete valuations • local search - chooses best neighbouring configuration • hill climbing neighbourhood = value of one variable changed • min-conflicts neighbourhood = value of selected conflicting variable changed • avoid local minimum => noise heuristics • random-walk sometimes picks neighbouring configuration randomly • tabu search few last configurations are forbidden for next step • does not guarantee completeness

5. Connectionist approach • Artificial Neural Networks • processors (cells) = <variable,value>on state means “value is assigned to the variable” • connections = inhibitory links between incompatible pairs • GENET starts from random configuration re-computes states using neighbouring cells till stable configuration found (equilibrium) learns violated constraints by strengthening weights • Incomplete (oscillation) 1 values 2 Z X Y variables

6. Constraint Optimisation • looking for best solution • quality of solution measured by application dependent objective function • Constraint Satisfaction Optimisation Problem • CSP • objective function: solution -> numerical valueNote: solution = complete labelling satisfying all the constraints • Branch & Bound (B&B) • the most widely used optimisation algorithm

7. Branch & Bound Constraint Optimisation • depth first search (like BT)+ under estimate of the objective function (minimisation)+ bound (initially set to plus infinity) • heuristic function:partial labelling -> under estimate of the objective function • pruning sub-tree under the partial labelling when • constraint inconsistency detected • heuristic value exceeds the bound • efficiency is determined by: • the quality of the heuristic function • whether a good bound is found early 

8. Over-Constrained Problems • What solution should be returned whenno solution exists? • impossible satisfaction of all constraints because of inconsistencyExample: X=5, X=4 • Solving methods • Partial CSP (PCSP) weakening original CSP • Constraint Hierarchies preferential constraints

9. Dressing Problem Over-Constrained Problems shirt: {red, white} footwear: {cordovans, sneakers} trousers: {blue, denim, grey} shirt x trousers: red-grey, white-blue, white-denim footwear x trousers: sneakers-denim, cordovans-grey shirt x footwear: white-cordovans red white shirt blue denim grey trousers footwear cordovans sneakers

10. Partial CSP Over-Constrained Problems • weakening a problem: • enlarging the domain of variable • enlarging the domain of constraint  • removing a variable • removing a constraint • one solution white - denim - sneakers shirt red white enlarged constraint’s domain blue denim grey footwear trousers cordovans sneakers

11. Partial CSP Over-Constrained Problems • Partial Constraint Satisfaction Problem • CSP • evaluation function: labelling -> numerical value • Task:find labelling optimal regarding the evaluation functionExample: maximising number of satisfied constraints • Usage: • over-constrained problems • optimisation problems (PCSP is a generalisation of CSOP) • obtaining “good enough” solution within fixed time

12. Constraint Hierarchies Over-Constrained Problems • constraints with preferences • solution respects the hierarchy • weaker constraints do not cause dissatisfaction of stronger constraint • shirt x trousers @ requiredfootwear x trousers @ strongshirt x footwear @ weak • two solutions red - grey - cordovans white - denim - sneakers shirt red white blue denim grey footwear trousers cordovans sneakers

13. Constraint Hierarchies Over-Constrained Problems • Hierarchy = (finite) set of labelled constraints levels Hi (H0 - required constraints, H1 - strongest non required …) • Solution: • S0={ | all required constraints are satisfied by  } • SH={ |  S0 s.t.    S0 better(,,H) }where better - comparator (partial ordering of valuations) • solving methods: • refining method (DeltaStar) • solve constraints from stronger to weaker levels • local propagation (DeltaBlue, SkyBlue) • repeatedly selects uniquely satisfiable constraints • planning + value propagation • hierarchies in optimisation problems • weak constraint objective function = optimum

14. Applications • assignment problems • stand allocation for airports • berth allocation to ships • personnel assignment • rosters for nurses • crew assignment to flights • network management and configuration • planning of cabling of telecommunication networks • optimal placement of base stations in wireless networks • molecular biology • DNA sequencing • analogue and digital circuit design

15. Scheduling Problems • the most successful application area • production scheduling (InSol Ltd.) • well-activity scheduling (Saga Petroleum) • forest treatment scheduling • planning production of jets (Dassault Aviation)

16. Advantages • declarative nature • focus on describing the problem to be solved, not on specifying how to solve it • co-operative problem solving • unified framework for integration of variety of special-purpose algorithms • semantic foundation • amazingly clean and elegant languages • roots in logic programming • applications • proven success

17. Limitations • NP-hard problems & tracktability • unpredictable behaviour • model stability • too high-level(new constraints, solvers, heuristics) • too low-level (modelling) • too local • non-incremental (rescheduling) • weak solver collaboration

18. Trends • modelling • global constraints (all_different) • modelling languages (Numerica, VisOpt) • understanding search • visualisation, performance debugging • hybrid algorithms • solver collaboration • parallelism • multi-agent technology

19. Resources • Conferences • Principles and Practice of Constraint Programming (CP) • The Practical Application of Constraint Technologies and Logic Programming (PACLP) • Journal • Constraints (Kluwer Academic Publishers) • Internet • Constraints Archivehttp://www.cs.unh.edu/ccc/archive • Guide to Constraint Programminghttp://kti.mff.cuni.cz/~bartak/constraints/

20. Constraint Satisfaction Problems

21. Quotations • “Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.” Eugene C. Freuder, Constraints, April 1997 • “Were you to ask me which programming paradigm is likely to gain most in commercial significance over the next 5 years I’d have to pick Constraint Logic Programming, even though it’s perhaps currently one of the least known and understood.” Dick Pountain, BYTE, February 1995

22. The Origins • Artificial Intelligence • Scene Labelling (Waltz) • Interactive Graphics • Sketchpad (Sutherland) • ThingLab (Borning) • Logic Programming • unification --> constraint solving • Operations Research • NP-hard combinatorial problems

23. Scene Labelling + + + + - - + + + + • first constraint satisfaction problem • Task:recognise objects in 3D scene by interpreting lines in 2D drawings • Waltz labelling algorithm • legal labels for junctions only • the edge has the same label at both ends

24. Interactive Graphics • Sketchpad (Sutherland) • ThingLab (Borning) • allow to draw and manipulate constrained geometric figures in the computer display

25. Outline • Constraint Satisfaction Problems (CSP) • Backtracking search for CSPs • Local search for CSPs CS 3243 - Constraint Satisfaction

26. What is CP? • CP = Constraint Programming • stating constraints about the problem variables • finding solution satisfying all the constraints • constraint = relation among several unknowns • Example: A+B=C, X>Y, N=length(S) … • Features: • express partial information X>2 • heterogeneous N=length(S) • non-directional X=Y+2: X Y+2  YX-2 • declarative manner “ • additive X>2,X<5  X<5,X>2 • rarely independent A+B=5, A-B=1

27. Constraint Satisfaction Problem • Consist of: • a set of variables X={x1,…,xn} • variables’ domains Di (finite set of possible values) • a set of constraints Example: • X::{1,2}, Y::{1,2}, Z::{1,2} • X = Y, X  Z, Y > Z • Solution of CSP • assignment of value from its domain to every variable satisfying all the constraints Example: • X=2, Y=2, Z=1

28. Constraint satisfaction problems (CSPs) • Standard search problem: • state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test • CSP: • state is defined by variablesXi with values from domainDi • goal test is a set of constraints specifying allowable combinations of values for subsets of variables • Simple example of a formal representation language • Allows useful general-purpose algorithms with more power than standard search algorithms

29. Constraint satisfaction problems (1) • Many problems can be formulated as finding an allocation of values to a set of variables, possibly optimal, that satisfies a set of constraints. • built a time table for the next semester with the minimum number of conflicts • schedule air crew assignements so that pilots don’t fly too many hours, have evenly-spaced breaks, … • find a 0/1 assignment to a propositional formula • 8-queen problem, cryptoarithmetic

30. Example: Map-Coloring • VariablesWA, NT, Q, NSW, V, SA, T • DomainsDi = {red,green,blue} • Constraints: adjacent regions must have different colors • e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)}

31. Example: Map-Coloring • Solutions are complete and consistent assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red,SA = blue,T = green CS 3243 - Constraint Satisfaction

32. Constraint graph • Binary CSP: each constraint relates two variables • Constraint graph: nodes are variables, arcs are constraints

33. Varieties of CSPs • Discrete variables • finite domains: • n variables, domain size d  O(dn) complete assignments • e.g., Boolean CSPs, incl.~Boolean satisfiability (NP-complete) • infinite domains: • integers, strings, etc. • e.g., job scheduling, variables are start/end days for each job • need a constraint language, e.g., StartJob1 + 5 ≤ StartJob3 • Continuous variables • e.g., start/end times for Hubble Space Telescope observations • linear constraints solvable in polynomial time by linear programming

34. Varieties of constraints • Unary constraints involve a single variable, • e.g., SA ≠ green • Binary constraints involve pairs of variables, • e.g., SA ≠ WA • Higher-order constraints involve 3 or more variables, • e.g., cryptarithmetic column constraints

35. Examples of CSPs F O R T Y + T E N + T E N S I X T Y 2 9 7 8 6 + 850 + 850 31486 Solution F=2, O=9 R=7, T=8 Y=6, E=5 N=0, I=1, X=4 • 8-queensproblem • Time-table schedule Mon Tue Wed Thu Fri 8 9 10 11

36. Example: Cryptarithmetic • Variables: F T U W R O X1 X2 X3 • Domains: {0,1,2,3,4,5,6,7,8,9} • Constraints: Alldiff (F,T,U,W,R,O) • O + O = R + 10 · X1 • X1 + W + W = U + 10 · X2 • X2 + T + T = O + 10 · X3 • X3 = F, T ≠ 0, F≠ 0 CS 3243 - Constraint Satisfaction

37. Real-world CSPs • Assignment problems • e.g., who teaches what class • Timetabling problems • e.g., which class is offered when and where? • Transportation scheduling • Factory scheduling • Notice that many real-world problems involve real-valued variables CS 3243 - Constraint Satisfaction

38. Constraint satisfaction problems (2) • Even though the domains are very different, the states can be described as partial assignment of values to variables. • The goal states satisfy each one of a set of constraints. • Exploit the structure of the problem!

39. Outline • Constraint Satisfaction Problems (CSP) • Backtracking search for CSPs • Local search for CSPs CS 3243 - Constraint Satisfaction

40. Systematic Search Methods • exploring the solution space • complete and sound • efficiency issues • Backtracking (BT) • Generate & Test (GT) exploring subspace exploringindividual assignments Z Y X

41. Generate & Test Systematic Search Methods • probably the most general problem solving method • Algorithm: generate labelling test satisfaction Drawbacks: Improvements: blind generator smart generator --> local search late discovery of testing within generator inconsistencies --> backtracking

42. Backtracking (BT) Systematic Search Methods • incrementally extends a partial solution towards a complete solution • Algorithm: assign value to variable check consistency until all variables labelled • Drawbacks: • thrashing • redundant work • late detection of conflict A 1 B 2 1 C 2 1 1 D 2 2 1 1 1  A = D, B  D, A+C < 4

43. GT & BT - Example Systematic Search Methods • Problem:X::{1,2}, Y::{1,2}, Z::{1,2}X = Y, X  Z, Y > Z generate & test backtracking

44. CSP search: Example S E N D + M O R E M O N E Y X1 X2X3 X4  M = 1 O = 0 N = 2 S 3 2 D 1 0 R 3 1 0 2 3 Y E = 3 Conflict!

45. CSP problem characteristics (1) • Special case of a search problem • Domains can be discrete or continuous • Commutativity: the order in which we apply the actions has no effect: at each node, consider only assignments to a single variable • During the search, once a constraint is violated, it remains so (monotonicity): stop and backtrack as soon as a constraint is violated • The order in which we pick variables and their values makes a big difference: we need smart heuristics

46. CSP problem characteristics (2) • The goal test is decomposed into a set of constraints on variables, rather than a single black box • When sets of variables are independent (no constraints between them) the problem is decomposable and subproblems can be solved independently. • A each step we must chech for consistency. We need constraint propagation methods!

47. Backtracking-Search: incremental • Search strategy: DFS 1. pick an uninstantiated variable, pick a value from its domain, check if some constraint is violated 2. if no constraint is violated, continue search recursively 3. else, backtrack: go back to previous decision and make another choice • Size of search tree: dn (number of leaves) • n, number of variables • d = max Di

48. Standard search formulation (incremental) Let's start with the straightforward approach, then fix it States are defined by the values assigned so far • Initial state: the empty assignment { } • Successor function: assign a value to an unassigned variable that does not conflict with current assignment  fail if no legal assignments • Goal test: the current assignment is complete • This is the same for all CSPs • Every solution appears at depth n with n variables use depth-first search • Path is irrelevant, so can also use complete-state formulation • b = (n - l )d at depth l, hence n! · dn leaves CS 3243 - Constraint Satisfaction

49. Backtracking search • Variable assignments are commutative}, i.e., [ WA = red then NT = green ] same as [ NT = green then WA = red ] • Only need to consider assignments to a single variable at each node  b = d and there are $d^n$ leaves • Depth-first search for CSPs with single-variable assignments is called backtracking search • Backtracking search is the basic uninformed algorithm for CSPs • Can solve n-queens for n≈ 25 CS 3243 - Constraint Satisfaction

50. Backtracking search CS 3243 - Constraint Satisfaction