Modelling with Constraints

Transcription

1 Masterclass Modelling with Constraints Part 1: Introduction Alan M Frisch Artificial Intelligence Group Dept of Computer Science University of York 12 December

2 Motivation A modern generation of solvers are powerful enough to solve problems that are of practical importance in many areas of computer science. Key properties of the solvers - off-the-shelf - black box - declarative input - many good ones are free for non-commercial use So, you may find these solvers useful. 2

3 Many Solvers, Many Input Languages Many types of solvers: - Constraint solvers (branch & prune; local search) - Boolean satisfiability (SAT) solvers (branch & prune; local search) - SAT modulo theory (SMT) solvers - Answer set programming (ASP) solvers - Mixed Integer Programming (MIP) solvers - Hybrids of the above Unfortunately, many input languages 3

4 Constraint Modelling Languages Originally developed as input languages to branch-and-prune constraint solvers. Variety of languages, but conceptually similar - MiniZinc, OPL, ESSENCE There are now translators from constraint modelling languages to a wide number of solvers - MiniZinc can be translated to about 10 solvers. 4

5 How to Solve a Problem Construct a constraint model of the problem - that is: reduce your problem to the finite-domain constraint satisfaction problem (FD-CSP) - that is: specify a set of constraints that every solution must satisfy Specify your model in some constraint modelling language. Input your model specification into a translator/solver. Translate solutions of the model to solutions of the original problem. 5

6 How to Solve a Problem Construct a constraint model of the problem - that is: reduce your problem to the finite-domain constraint satisfaction problem (FD-CSP) - that is: specify a set of constraints that every solution must satisfy Specify your model in some constraint modelling language. Input your model specification into a translator/solver. Translate solutions of the model to solutions of the original problem. 5

7 How to Solve a Problem Construct a constraint model of the problem - that is: reduce your problem to the finite-domain constraint satisfaction problem (FD-CSP) - that is: specify a set of constraints that every solution must satisfy This is constraint modelling We will focus on modelling combinatorial (decision and optimisation) problems 6

8 A Short Introduction to Modelling with Constraints 7

9 What is the Finite-Domain CSP? An instance of the finite-domain CSP consists of: - A finite set V of variables - A domain D, a function that maps every variable to a finite set - A finite set of constraints, each restricting the values that some subset of variables can take. A solution to an instance of FD-CSP is an assignment that maps every variable v in V to an element of its domain, D(v), such that each constraint is satisfied. 8

10 What is the Finite-Domain CSP? Given an instance I of FD-CSP the goal is either - determine if I is satisfiable, or - find a solution of I, or - find all solutions of I, or - find the best solution according to some given objective. 9

11 Modelling 4-Queens Problem Place 4 queens on a 4 x 4 chessboard so that no queen attacks another. Two solutions: Note: No two queens are in the same column, same upward diagonal or downward diagonal. 10

12 Modelling 4-Queens Problem A B C D 11

13 A B C D Modelling 4-Queens Problem Variables: A, B, C, D 11

14 A B C D Modelling 4-Queens Problem Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} 11

15 A B C D Modelling 4-Queens Problem Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} A B A and B are in different columns A +1 B A and B are in different downward diagonals A -1 B A and B are in different upward diagonals similar for B and C similar for C and D A C A and C are in different columns A + 2 C A and C are in different downward diagonals A - 2 C A and C are in different upward diagonals similar for B and D A D A and D are in different columns A + 3 D A and D are in different downward diagonals A - 3 D A and D are in different upward diagonals 11

16 Solving 4-Queens Model with Backtracking Note: Variable ordering affects search space. Value ordering affects how it is explored. 12

17 Solving 4-Queens Model with MAC 13

18 Solving 4-Queens Model with MAC C and D are in different downward diagonals 14

19 Solving 4-Queens Model with MAC B and C are in different upward diagonals 15

20 Solving 4-Queens Model with MAC C and D are in different columns 16

21 Solving 4-Queens Model with MAC B and D are in different columns 17

22 Solving 4-Queens Model with MAC 18

23 A B C D Remodelling 4-Queens Problem Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} A B A +1 B NoAttack(A,B,1) A -1 B similar for B and C NoAttack(B,C,1) similar for C and D NoAttack(C,D,1) A C A + 2 C NoAttack(A,C,2) A - 2 C similar for B and D NoAttack(B,D,2) A D A + 3 D NoAttack(A,D,3) A - 3 D 19

24 Solving New 4-Queens Model with MAC noattack(a,b,1) noattack(c,d,1) 20

25 Adding a Symmetry-Breaking Constraint to 4-Queens Model A B C D Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} NoAttack(A,B,1) NoAttack(B,C,1) NoAttack(C,D,1) NoAttack(A,C,2) NoAttack(B,D,2) NoAttack(A,D,3) 21

26 Adding a Symmetry-Breaking Constraint to 4-Queens Model A B C D Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} NoAttack(A,B,1) NoAttack(B,C,1) NoAttack(C,D,1) NoAttack(A,C,2) NoAttack(B,D,2) NoAttack(A,D,3) A 2 21

27 Adding a Symmetry-Breaking Constraint to 4-Queens Model A B C D Variables: A, B, C, D D(A) = D(B) = D(C)= D(D) = {1,2,3,4} NoAttack(A,B,1) NoAttack(B,C,1) NoAttack(C,D,1) NoAttack(A,C,2) NoAttack(B,D,2) NoAttack(A,D,3) A 2 21

28 Modelling the n-queens Problem Class Parameters: n Vars: Q[1..n] Domain: D(Q[i]) = {1..n} (1 i n) Constraints: i 1..n-1. j i+1..n. noattack(q[i],q[j],j-i) 22

29 Modelling the n-queens Problem Class Parameters: n Vars: Q[1..n] Domain: D(Q[i]) = {1..n} (1 i n) Constraints: i 1..n-1. j i+1..n. noattack(q[i],q[j],j-i) This is just a way of writing a set of constraints, namely { noattack((q[i],q[j],j-i) i 1..n-1, j I+1..n } 22

30 Modelling the n-queens Problem Class Parameters: n Vars: Q[1..n] Domain: D(Q[i]) = {1..n} (1 i n) Constraints: i 1..n-1. j i+1..n. noattack(q[i],q[j],j-i) 23

31 Modelling the n-queens Problem Class Parameters: n Vars: Q[1..n] Domain: D(Q[i]) = {1..n} (1 i n) Constraints: i 1..n-1. j i+1..n. noattack(q[i],q[j],j-i) I call any language of this form a Declarative Constraint Language 23

32 Modelling the n-queens Problem Class in a Procedural Language with a Constraint Library procedure Queens(n:int) declare var Q[1..n]: 1..n; for i := 1..n-1 do for j := i+1..n do post(noattack(q[i],q[j],j-i)); search(q); end; 24

33 Constraint Modelling: Morals of the Story Some models of a problem can be solved much faster than others. Adding implied constraints to a model sometimes (not always) reduces the size of the search space and never increases it. Adding symmetry-breaking constraints reduces the size of the search space. 25