Constraint Satisfaction

1. Constraint Satisfaction taking advantage of internal structure of states when paths don’t matter

2. Constraint satisfaction problems (CSPs) • no prescribed start state • path does not matter • acceptable states must satisfy constraints • functions of state variables • among acceptable states, goal states will optimize an objective function D Goforth - COSC 4117, fall 2003

3. Example problem – timetable of university (1) • state - collection of schedule objects • schedule • professor(~102) (~350 at LU) • course (~103) • time slot (~10) (16 at LU) • location (~102) • constraints • objective function D Goforth - COSC 4117, fall 2003

4. Example problem – timetable of university (2) • constraints • professors assigned to courses • one course per location per time slot • location capacity >= course (projected) enrollment • professors assigned only once per time slot • rooms assigned only once per time slot • courses in program in different time slots D Goforth - COSC 4117, fall 2003

5. Example problem – timetable of university (3) • objective function • minimize total course enrollment in early morning time slots • distribute program courses across days • concentrate professor assignments on same day • accommodate special requests D Goforth - COSC 4117, fall 2003

6. Solving CSPs • analysis of problem state structure is critical • two basic approaches: • start state is null: assign state variables one at a time and backtrack if constraint is violated • assign variables ‘randomly/intelligently’ to make a start state and use greedy search by changing variable assignments D Goforth - COSC 4117, fall 2003

7. Example problem – timetable of university (4) • analyzing the problem • e.g. fix time slots for all weekly sessions of a course (tutorials?) • problem solving approaches • allocate courses to rooms and times one after the other • assign all courses then start moving those involved in constraint violations D Goforth - COSC 4117, fall 2003

8. CSP – problem definition • state representation (R&N) • state variables X1, X2, …, Xn • Xi from domain Di • constraints C1, C2, …, Cm • Cj are functions of Xi (unary, binary, … fns) • solution – assignment of values vi to all Xi so all constraint functions Cj are satisfied • objective function – optimized function of Xi over solution set D Goforth - COSC 4117, fall 2003

9. Example problem – timetable of university (4) • variables: array of assignments of courses (ck) to locations (loci)and time (tj) slots • ck:prof,enroll,prog • loci: capacity • tj: time, day • A[loc,t] = ck • constraints: • loc.capacity >= c.enroll (unary) • if (loci1 != loci2) A[loci1,tj].prof != A[loci2,tj].prof D Goforth - COSC 4117, fall 2003

10. Modelling a state as constraint graph • variables are nodes loc 1: 25 loc 2: 100 c 1: Ann 20 c 2: Ben 45 c 3: Cec 15 c 4: Ben 80 c 5: Ann 90 t 1 t 2 t 3 D Goforth - COSC 4117, fall 2003

11. Modelling a state as constraint graph • unary constraints reduce domains for A(loc1, t) {c1,c3} for A(loc2, t) {c1,c2,c3,c4,c5} loc 1: 25 loc 2: 100 c 1: Ann 20 c 2: Ben 45 c 3: Cec 15 c 4: Ben 80 c 5: Ann 90 t 1 t 2 t 3 D Goforth - COSC 4117, fall 2003

12. Modelling a state as constraint graph • binary constraints are edges for A(loc1, t) {c1,c3} for A(loc2, t) {c1,c2,c3,c4,c5} loc 1: 25 loc 2: 100 c 1: Ann 20 c 2: Ben 45 c 3: Cec 15 c 4: Ben 80 c 5: Ann 90 t 1 t 2 A[loc1,tj].prof != A[loc2,tj].prof t 3 D Goforth - COSC 4117, fall 2003

13. Modelling a state as constraint graph • multiple constraints are ‘hubs’ for A(loc1, t) {c1,c3} for A(loc2, t) {c1,c2,c3,c4,c5} loc 1: 25 loc 2: 100 c 1: Ann 20 c 2: Ben 45 c 3: Cec 15 c 4: Ben 80 c 5: Ann 90 t 1 t 2 each course in one slot only t 3 D Goforth - COSC 4117, fall 2003

14. Variable domains • discrete finite • discrete infinite • continuous Domains may be distinct or shared among variables (typical) D Goforth - COSC 4117, fall 2003

15. Constraints • enumerations of possible combinations • functions of variables • linear, non-linear, … • unary, binary,… D Goforth - COSC 4117, fall 2003

16. Solution by incremental formulation • initial state – no variable assigned • successor function for transition: • pick a variable and set a value that does not conflict with previous assignments by applying constraint functions • goal – all variables assigned D Goforth - COSC 4117, fall 2003

17. Solution by incremental formulation • advantage • depth of search is limited to number of variables in state • depth limited dfs is natural algorithm • ‘backtracking search’ D Goforth - COSC 4117, fall 2003

18. c 1 c 5 c 5 c 5 c 5 c 5 c 1 c 5 c 1: Ann 20 c 2: Ben 45 c 3: Cec 15 c 4: Ben 80 c 5: Ann 90 Backtracking search loc 1: 25 loc 2: 100

19. Efficient Backtracking search (1) • Forward checking - Propagate constraints • when variable is assigned, use constraints to reduce value sets of remaining variables • e.g. when course is assigned: • remove course from domain of all remaining variables • remove courses with same prof from allocations in same time slot D Goforth - COSC 4117, fall 2003

20. Efficient Backtracking search (2) • Variable ordering – which variable to assign a value next? • MRV heuristic – pick most constrained variable (minimum remaining values) • e.g. – book small room first • degree heuristic – pick variable involved in most constraints (propagates and prunes most) D Goforth - COSC 4117, fall 2003

21. Efficient Backtracking search (3) • Value ordering: once variable is selected, which value to try first? • least constraining value heuristic – pick value that reduces value sets of remaining variables the least • e.g. when assigning a course: • pick location with smallest capacity that will accommodate the course (risk?) D Goforth - COSC 4117, fall 2003

22. Efficient Backtracking search (4) • Arc consistency: stronger constraint propagation • check for consistency between variables after value is set forward checking remove variables ? arc consistency D Goforth - COSC 4117, fall 2003

23. Efficient Backtracking search (5) • k-consistency: extending arc-consistency forward checking remove variables arc consistency D Goforth - COSC 4117, fall 2003

24. Norvig and Russell example chapter04b.ps D Goforth - COSC 4117, fall 2003

25. DFS ‘chronological’ backtracking • when search fails X6 (no value can be set for variable) • back up one level, try again v5v5 • useless IF no constraint between X5, X6 X1 X2 X3 X4 X5 X6 D Goforth - COSC 4117, fall 2003

26. DFS backjumping • when search fails X6 (no value can be set for variable) • look at conflict set of X6 {X3, X1} • jump back to change v3 v3 X1 X2 X3 X4 X5 X6 D Goforth - COSC 4117, fall 2003

27. Solution by local search • start state has values for every variable • state variables X1=v1, X2=v2, …, Xn=vn • constraints C1, C2, …, Cm • some satisfied, some violated • objective function not optimized • successor function changes one or more variables – evaluate to minimize conflicts D Goforth - COSC 4117, fall 2003

28. Solution by local search • timetable example: • start state – assign all courses to room/time slot • change assignments to reduce conflicts D Goforth - COSC 4117, fall 2003

29. advantages of local search • solves some problems very fast • flexible in online situations – revised conditions can be re-solved with minimal changes • e.g. course enrollment projections turn out to be wrong – conflicts with room sizes • takes advantage of any known partial solutions installed in start state D Goforth - COSC 4117, fall 2003

30. Constraint graph structure • independent variables (unary constraints only) easy but rare • tree graphs – process variables in top-down order from any node as root • general graphs – may be reducible (by removing and assigning some nodes) to trees • general graphs – may be reducible by clustering into subgraphs D Goforth - COSC 4117, fall 2003

31. Examples • crossword puzzle construction • p.158 #5.4 • cryptarithmetic puzzle – section 5.2 • other example • Sudoku puzzle D Goforth - COSC 4117, fall 2003

32. cB3 1 2 3 4 5 6 7 8 9 y 1 2 3 4 5 6 7 8 9 x|A B C D E F G H I Sudoku constraints From rules Constraints on all cells: For rows, columns, squares Every integer occurs No integer is repeated From data (particular game) Constraints on some cells Specific value Underconstrained – many solutions Overconstrained – no solutions