Examples for Discrete Constraint ProgrammingBelaid MOA UVIC, SHRINC Project

Examples

Map coloring Problem Cryptarithmetic N-queens problem Magic sequence Magic square Zebra puzzle Uzbekian puzzle A tiny transportation problem Knapsack problem graceful labeling problem

Map Coloring Problem (MCP)

Given a map and given a set of colors, the problem is how to color the map so that the regions sharing a boundary line dont have the same color.

Map Coloring Problem (MCP)

MCP can be viewed as a graph coloring problem:a b c b

ad d c

Modeling MCP

MCP can be Modeled as a CSP: A set of variables representing the color of each region The domain of the variables is the set of the colors used. A set of constraints expressing the fact that countries that share the same boundary are not colored with the same color.

Modeling (MCP)

Example - Australia Variables = {a,b,c,d,e,f,g} Domain = {red, blue, green} Constraints: a!=b, a!=c b!=c, b!=d c!=e, c!=f e!=d, e!=f

b d a c e f

g

Coding MCP

OPL studio - Ilog Solver: 54 solutions found

enum Country={a,b,c,d,e,f,g}; enum Color = {red, green, blue};var Color color[Country]; solve{ color[a]color[b];color[a]color[c]; color[b]color[c];color[b]color[d]; color[c]color[e];color[c]color[f]; color[e]color[d];color[e]color[f]; color[b]color[c];color[b]color[d]; };

Coding MCP

Coding MCP

ECLiPSe

:- lib(fd). coloured(Countries) :Countries=[A,B,C,D,E,F,G], Countries :: [red,green,blue], ne(A,B), ne(A,C), ne(B,C), ne(B,D), ne(C,E), ne(C,F), ne(E,D), ne(E,F), labeling(Countries).ne(X,Y) :- X##Y.

Coding MCP

Cryptarithmetic

The problem is to find the digits corresponding to the letters involved in the following puzzle: SEND + MORE MONEY

Cryptarithmetic Modeling

A CSP model for Cryptarithmetic problem: Variables={S,E,N,D,M,O,R,Y} Domain={0,1,2,3,4,5,6,7,8,9} Constraints The variables are all different S!=0, M!=0 3 2 S*10 +E*10 +N*10+D {SEND} + M*103+O*102+R*10+E {MORE}= M*104+O*103+N*102+E*10+Y {MONEY}

Coding Cryptarithmetic

OPL Studio - ILog Solver

enum letter = {S,E,N,D,M,O,R,Y}; range Digit 0..9; var Digit value[letter]; solve{ alldifferent(value); value[S]0; value[M] 0; value[S]*1000+value[E]*100+value[N]*10+value[D]+ value[M]*1000+value[O]*100+value[R]*10+value[E] = value[M]*10000+value[O]*1000+value[N]*100+value[E]*10+value[Y] };

Coding Cryptarithmetic

Coding Cryptarithmetic

ECLiPSe

:- lib(fd). sendmore(Digits) :Digits = [S,E,N,D,M,O,R,Y], Digits :: [0..9], alldifferent(Digits), S #\= 0, M #\= 0, 1000*S + 100*E + 10*N + D + 1000*M + 100*O + 10*R + E #= 10000*M + 1000*O + 100*N + 10*E + Y,labeling(Digits).

Coding Cryptarithmetic

N-Queens Problem

The problem is to put N queens on a board of NxN such that no queen attacks any other queen.

A queen moves vertically, horizontally and diagonally

Modeling N-Queens Problem

The queens problem can be modeling via the following CSP Variables={Q1,Q2,Q3,Q4,...,QN}. Domain={1,2,3,,N} represents the column in which the variables can be. Constraints

Queens not on the same row: already taken care off by the good modeling of the variables. Queens not on the same column: Qi != Qj Queens not on the same diagonal: |Qi-Qj| != |i-j|

Coding N-Queens Problem

OPL Studio - ILog Solverint n