Constraint Satisfaction Problems

Transcription

1 Constraint Satisfaction Problems Adrian Groza Department of Computer Science Technical University of Cluj-Napoca 12 Nov 2014

2 Outline 1 Constraint Reasoning 2 Systematic Search Methods Improving backtracking efficiency

3 Example Constraints programming represents one of the closest approaches computer science has made to the Holy Grail of programming: the user states the problem, the computer solves it. Eugene C. Freuder, Inaugural issue of the Constraints Journal, Place numbers 1 through 8 on nodes 1 each number appears exactly once 2 no connected nodes have consecutive numbers

4 Heuristic Search To succeed, try first where you are most likely to fail. Deal with hard cases first: they can only get more difficult if you put them off. Which nodes are hardest to number? Guess a value, but be prepared to backtrack Which are the least constraining values to use? Symmetry means we don t need to consider: 8 1

5 Inference & constraint propagation I 1 each number appears exactly once 2 no connected nodes have consecutive numbers We can now eliminate many values for other nodes

6 Inference & constraint propagation II 1 each number appears exactly once 2 no connected nodes have consecutive numbers We can now eliminate many values for other nodes

7 Inference & constraint propagation III 1 each number appears exactly once 2 no connected nodes have consecutive numbers By simmetry

8 Inference & constraint propagation IV 1 each number appears exactly once 2 no connected nodes have consecutive numbers

9 Inference & constraint propagation V 1 each number appears exactly once 2 no connected nodes have consecutive numbers By symmetry

10 Inference & constraint propagation VI 1 each number appears exactly once 2 no connected nodes have consecutive numbers

11 Inference & constraint propagation VII 1 each number appears exactly once 2 no connected nodes have consecutive numbers

12 Inference & constraint propagation VIII 1 each number appears exactly once 2 no connected nodes have consecutive numbers And propagate

13 Inference & constraint propagation IX 1 each number appears exactly once 2 no connected nodes have consecutive numbers Guess a value, but be prepared to backtrack

14 Inference & constraint propagation X 1 each number appears exactly once 2 no connected nodes have consecutive numbers And propagate

15 Inference & constraint propagation XI 1 each number appears exactly once 2 no connected nodes have consecutive numbers More propagation?

16 Inference & constraint propagation XII 1 each number appears exactly once 2 no connected nodes have consecutive numbers

17 Inference & constraint propagation XIII 1 each number appears exactly once 2 no connected nodes have consecutive numbers A solution

18 Constraint programming methodology 1 Model problem specify in terms of constraints on acceptable solutions: constraint satisfaction problem define/choose constraint model: variables, domains, constraints 2 Solve model define/choose algorithm define/choose heuristics 3 Verify and analyze solution

19 Constraints Properties A logical relation among several unknowns (variables) May specify partial information: X > 2, the circle is inside the square Non-directional: two variables X, Y can be used to infer a constraint on X given a constraint on Y and vice versa: X=Y+2 Declarative: specify what relationship must hold without specifying a computational procedure to enforce that relationship Additive: the order of imposition of constraints does not matter, all that matters, at the end is that the conjunction of constraints is in effect Rarely independent: typically constraints in the constraint store share variables.

20 Constraint satisfaction problem (CSP) A CSP is defined by a set of variables: X, Y, Z,... a domain of values for each variable: X :: {1, 2}, Y :: {1, 2}, Z :: {1, 2} a set of constraints between variables: X = Y, X Z, Y > Z A solution is an assignment of a value to each variable that satisfies the constraints: X = 2, Y = 2, Z = 1 Given a CSP Determine whether it has a solution or not (satisfiability) Find any solution Find all solutions Find an optimal solution, given some cost function

21 Give me some examples Puzzle Map coloring N-queen problem Sudoku Cryptarithmetic problem

22 Constraint Model For Puzzle variables v 1,..., v 8 domains {1,..., 8} constraints v 1 v 2 1 v 1 v 3 1. v 7 v 8 1 alldifferent(v 1,..., v 8 )

23 Constraint Model For Graph Coloring Given k colors, does there exist a coloring of the nodes such that adjacent nodes are assigned different colors variables v 1, v 2, v 3, v 4, v 5 domains {yellow, grey, blue} constraints v i v j if v i and v j are adjacent

24 Constraint Model For N-queens Place n-queens on an n x n board so that no pair of queens attacks each other variables x 1, x 2, x 3, x 4 domains {1, 2, 3, 4} constraints x i x j and x i x j i j a solution x 1 2, x 2 4, x 3 1, x 4 3

25 Modern Art Accident

26 Crypto-arithmetic problem Constraints [S,E,N,D,M,O,R,Y] in 0..9 alldifferent([s,e,n,d,m,o,r,y]) 1000*S + 100*E + 10*N + D+1000*M + 100*O + 10*R + E = 10000*M *O + 100*N + 10*E + Y More Constraints? S 0, M 0 M = 1, S = 9, O = 0, N + R > 10 N = E + 1, E in 2..7 N in 3..8, D, R, Y in 2..8 R in 3..8 N + R = 1E E < 7 R = 8 from N + R > 10, N = E + 1 E in 2..6, N in 3..7

27 Crypto-arithmetic problem E+D = Y+10*C1 C1+N+R = E+10*C2 C2+E+O = N+10*C3 C3+S+M = 10*M+O E,N,D,O,R,Y::{0,..,9} S,M::{1,..,9} C1,C2,C3::{0,1}

28 Send Most Money SEND+MOST=MONEY Money should be maximal

29 Example problem: Job-shop scheduling I Car assembly - 15 tasks: install axles (front and back), affix all four wheels (right and left, front and back), tighten nuts for each wheel, affix hubcaps, and inspect the final assembly, in maximum 30 minutes. Variables X = {Axle F, Axle B, Wheel RF, Wheel LF, Wheel RB, Wheel LB, Nuts RF, Nuts LF, Nuts RB, Nuts LB, Cap RF, Cap LF, Cap RB, Cap LB, Inspect} Precedence constraints: T 1 + d1 T 2 Axles have to be in place before the wheels are put on, and it takes 10 minutes to install an axle, Axle F + 10 WheelRF Axle B + 10 WheelRB Axle F + 10 WheelLF Axle B + 10 WheelLB

30 Example problem: Job-shop scheduling II For each wheel, we must affix the wheel (which takes 1 minute), then tighten the nuts (2 minutes) Wheel RF + 1 Nuts RF Nuts RF + 2 Cap RF Wheel LF + 1 Nuts LF Nuts LF + 2 Cap LF Wheel RB + 1 Nuts RB Nuts RB + 2 Cap RB Wheel LB + 1 Nuts LB Nuts LB + 2 Cap LB 4 our workers to install wheels, but they have to share one tool that helps put the axle in place. Disjunctive constraint to say that Axle F and Axle B must not overlap in time: (Axle F + 10 Axle B ) or (Axle B + 10 Axle F ) The inspection comes last and takes 3 minutes: X, X Inspect, X + d X Inspect All assembly done in 30 minute: D i = {1, 2, 3,..., 27}

31 Alldiff (A1,A2,A3,A4,A5,A6, A7, A8, A9) Alldiff (B1,B2,B3,B4,B5,B6,B7,B8,B9)... Alldiff (A1,B1,C1,D1,E1, F1,G1,H1, I1) Alldiff (A2,B2,C2,D2,E2, F2,G2,H2, I2)... Global constraints Involves an arbitrary number of constraints (but not necessarily all variables).

32 Global constraints At most contraint- resource contraints Let P1,..., P4 numbers of personnel assigned to each of four tasks. No more than 10 personnel are assigned in total: Atmost(10, P1, P2, P3, P4). Assume D = {3, 4, 5, 6} - detect inconsistency by checking the sum of the minimum values of the current domains; Assume D = {2, 3, 4, 5, 6} enforce consistency by deleting the maximum value of any domain if it is not consistent with the minimum values of the other domains Bounds propagation Example (Airline-scheduling problem) Two flights, F 1 and F 2, for which the planes have capacities 165 and 385, respectively: D 1 = [0, 165] and D 2 = [0, 385]. F 1 + F 2 = 420 D 1 = [35, 165] and D 2 = [255, 385]

33 Outline 1 Constraint Reasoning 2 Systematic Search Methods Improving backtracking efficiency

34 Systematic Search Methods Constraints, an Ultimate Anti NP-Hard weapon? Most problems that the constraint programming concerns belong to the group that conventional programming techniques find hardest Example Schedule the loading and unloading of 10 ships using only 5 berths (about 10 million) alternatives in the worst case: 3hours 10 berths and 20 ships: 300 million years No need to explore all alternatives: Some berths are too small for some ships Cannot load two ships in the same berth, at the same time You don t need an optimal schedule: break the problem in two parts - 6 hours

35 Systematic Search Methods Systematic Search Methods exploring the solution space complete and sound Complete- guarantees finding a solution or proving its non-existence Sound- guarantees finding only valid solutions efficiency issues Generate and Test (GT) Backtracking (BT)

36 Systematic Search Methods Systematic Search Methods Problem: X :: {1, 2}, Y :: {1, 2}, Z :: {1, 2}, X = Y, X Z, Y > Z Generate and Test Probably the most general problem solving method Algorithm: 1 generate labelling 2 test satisfaction Drawbacks: blind generator, late Backtracking The basic uninformed algorithm for CSP. Incrementally extends a partial solution towards a complete solution Algorithm: Build a partial solution: a partial consistent assignment Extend consistently the partial solution: one new assigned variable each time

37 Improving backtracking efficiency 1 Which variable should be assigned next? 2 In what order should its values be tried? 3 Can we detect inevitable failure early? 4 Can we take advantage of problem structure?

38 Improving backtracking efficiency Minimum remaining values Choose the variable with the fewest legal values (or most constrained variable or fail-first heuristic)

39 Improving backtracking efficiency Degree heuristic Choose the variable with the most constraints on remaining variables

40 Improving backtracking efficiency Least constraining value Given a variable, choose the least constraining value: the one that rules out the fewest values in the remaining variables Combining these heuristics makes 1000 queens feasible

41 Improving backtracking efficiency Constraint propagation Arc consistency : X Y is consistent iff for every value x of X there is some allowed y

42 Improving backtracking efficiency Constraint propagation Arc consistency : X Y is consistent iff for every value x of X there is some allowed y

43 Improving backtracking efficiency Weaknesses of Backtracking Thrashing: throws away the reason of the conflict Example: A, B, C, D, E :: 1..10, A > E: tries all the assignments for B,C,D before finding that A 1 The chronological backtracking backtracks to the Queen 5 and it finds another column for this queen (column H). However, it is still impossible to place the Queen 6 Solution: backjumping (jump to the source of the failure) The closest queen that can resolve the conflict is Queen 4 because then there is a chance that column D can be used for Queen 6.

44 Improving backtracking efficiency Weaknesses of Bactkracking Redundant Work Unnecessary constraint checks are repeated Even if the conflicting values of variables they are not remembered

45 Improving backtracking efficiency Weaknesses of Backtracking Late detection of conflict Constraint violation is discovered only when the values are known Example: A, B, C, D, E :: 1..10, A = 3 E The fact that A > 2 is discovered when labelling E Solution: forward checking - forward check of constraints

46 Improving backtracking efficiency Forward Checking

47 Improving backtracking efficiency Local Search for CSP

48 Improving backtracking efficiency 1 Which variable should be assigned next? 2 In what order should its values be tried? 3 Can we detect inevitable failure early? 4 Can we take advantage of problem structure?

49 Improving backtracking efficiency