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

D Goforth - COSC 4117, fall 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 Example problem – timetable of university (1)  state - collection of schedule objects schedule  professor(~10 2 ) (~350 at LU)  course (~10 3 )  time slot (~10) (16 at LU)  location (~10 2 )  constraints  objective function

D Goforth - COSC 4117, fall 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 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 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 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 CSP – problem definition  state representation (R&N) state variables X 1, X 2, …, X n  X i from domain D i constraints C 1, C 2, …, C m  Cj are functions of Xi (unary, binary, … fns) solution – assignment of values v i to all X i so all constraint functions C j are satisfied objective function – optimized function of X i over solution set

D Goforth - COSC 4117, fall Example problem – timetable of university (4)  variables: array of assignments of courses (c k ) to locations (loc i )and time (t j ) slots c k :prof,enroll,prog loc i : capacity t j : time, day A[loc,t] = c k  constraints: loc.capacity >= c.enroll (unary) if (loc i1 != loc i2 ) A[loc i1,t j ].prof != A[loc i2,t j ].prof

D Goforth - COSC 4117, fall Modelling a state as constraint graph  variables are nodes loc 1: 25loc 2: 100 t 1 t 2 t 3 c 1:Ann20 c 2:Ben45 c 3:Cec 15 c 4:Ben80 c 5:Ann90

D Goforth - COSC 4117, fall Modelling a state as constraint graph  unary constraints reduce domains loc 1: 25loc 2: 100 t 1 t 2 t 3 c 1:Ann20 c 2:Ben45 c 3:Cec 15 c 4:Ben80 c 5:Ann90 for A(loc 1, t) {c 1,c 3 } for A(loc 2, t) {c 1,c 2,c 3,c 4,c 5 }

D Goforth - COSC 4117, fall Modelling a state as constraint graph  binary constraints are edges loc 1: 25loc 2: 100 t 1 t 2 t 3 c 1:Ann20 c 2:Ben45 c 3:Cec 15 c 4:Ben80 c 5:Ann90 for A(loc 1, t) {c 1,c 3 } for A(loc 2, t) {c 1,c 2,c 3,c 4,c 5 } A[loc 1,t j ].prof != A[loc 2,t j ].prof

D Goforth - COSC 4117, fall Modelling a state as constraint graph  multiple constraints are ‘hubs’ loc 1: 25loc 2: 100 t 1 t 2 t 3 c 1:Ann20 c 2:Ben45 c 3:Cec 15 c 4:Ben80 c 5:Ann90 for A(loc 1, t) {c 1,c 3 } for A(loc 2, t) {c 1,c 2,c 3,c 4,c 5 } each course in one slot only

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

D Goforth - COSC 4117, fall Constraints  enumerations of possible combinations  functions of variables linear, non-linear, … unary, binary,…

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

Backtracking search loc 1: 25 loc 2: 100 c 1:Ann20 c 2:Ben45 c 3:Cec 15 c 4:Ben80 c 5:Ann90 c 5c 1c 5 c 1 c 5

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

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

D Goforth - COSC 4117, fall DFS ‘chronological’ backtracking  when search fails X6 (no value can be set for variable)  back up one level, try again v 5 v 5  useless IF no constraint between X 5, X 6 X1 X2 X3 X4 X5 X6

D Goforth - COSC 4117, fall 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 Solution by local search  start state has values for every variable state variables X 1 =v 1, X 2 =v 2, …, X n =v n constraints C 1, C 2, …, C m  some satisfied, some violated objective function not optimized  successor function changes one or more variables – evaluate to minimize conflicts

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

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 y y x|A B C D E F G H I c B3