Examples for Discrete
Constraint Programming
Belaid MOA
UVIC, SHRINC Project
Examples

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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 don’t have
the same color.
Map Coloring Problem (MCP)

MCP can be viewed as a graph coloring
problem:
a
b
c
b
a
d
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 Solver
int n << "number of queens:";
range Domain 1..n;
var Domain queens[Domain];
solve {
alldifferent(queens);
forall(ordered i,j in Domain) {
abs(queens[i]-queens[j])<> abs(i-j) ;
};
};
Coding N-Queens Problem
Coding N-Queens Problem

ECLiPSe
:-lib(fd).
nqueens(N, Q):length(Q,N),
Q::1..N,
alldifferent(Q),
( fromto(Q, [Q1|Cols], Cols, []) do
( foreach(Q2, Cols), param(Q1), count(Dist,1,_) do
Q2 - Q1 #\= Dist, Q1 - Q2 #\= Dist
)
),
search(Q).
search([]).
search(Q):deleteff(Var,Q,R),
indomain(Var), search(R).
Coding N-Queens Problem
Magic sequence problem


Given a finite integer n, the problem consists of finding a
sequence S = (s0,s1,…,sn), such that si represents the number
of occurrences of i in S.
Example:
 (2, 0, 2, 0)
 (1,2,1,0)
Modeling MSP

MSP can be modeled by the following CSP
 variables: s0,s1, …,sn-1
 Domain:{0,1,…,n}
 constraints:


number of occurences of i in (s0,s1, …,sn-1) is si.
Redundant constraints:


the sum of s0,s1, …,sn-1 is n
the sum of i*Si, i in [0..n-1] is n
Coding MSP

OPL - ILog Solver
int n << "Number of Variables:";
range Range 0..n-1;
range Domain 0..n;
int value[i in Range ] = i;
var Domain s[Range];
solve {
distribute(s,value,s); //global constraint s[i]=sum(j in Range)(s[j]=i)
sum(i in Range) s[i] = n; //redundant constraint
sum(i in Range) s[i]*i = n; //redundant constraint
};
Coding MSP

ECLiPSe
:- lib(fd).
:- lib(fd_global).
:- lib(fd_search).
solve(N, Sequence) :length(Sequence, N),
Sequence :: 0..N-1,
( for(I,0,N-1),
foreach(Xi, Sequence),
foreach(I, Range), param(Sequence) do
occurrences(I, Sequence, Xi)
),
N #= sum(Sequence), % two redundant constraints
N #= Sequence*Range,
search(Sequence, 0, first_fail, indomain, complete, []).%search procedure
Magic square problem


A magic square of size N is an NxN square filled with
the numbers from 1 to N2 such that the sums of each
row, each column and the two main diagonals are
equal.
Example:
Modeling Magic square pb

Magic square problem can be viewed as a CSP with the
following properties:

Variables: the elements of the matrix representing the
square

Domain: 1..N*N

Constraints:

magic sum = sum of the columns = sum of the rows =
sum of the down diagonal = sum of the up diagonal

Remove symmetries

Redundant constraint:

magic sum = N(N2+1)/2
Coding Magic square pb

OPL Studio:
int N << "The length of the square:";
range Dimension 1..N;
range Values 1..N*N;
int msum = N*(N*N+1)/2; //magic sum
var Values square[1..N,1..N];
solve {
//The elements of the square are all different
alldifferent(square);
//the sum of the diagonal is magic sum
msum = sum(i in Dimension)square[i,i];
//the sum of the up diagonal is magic sum
Coding Magic square pb
msum = sum(i in Dimension)square[i,N-i+1];
//the sum of the rows and columns are all magic sum
forall(i in Dimension){
msum = sum(j in Dimension)square[i,j];
msum = sum(k in Dimension)square[k,i];
};
//remove symmetric behavior of the square
square[N,1] < square[1,N];
square[1,1] < square[1,N];
square[1,1] < square[N,N];
square[1,1] < square[N,1];
};
Magic square problem
Coding Magic square pb

ECLiPSe:
:- lib(fd).
magic(N) :Max is N*N,
Magicsum is N*(Max+1)//2,
dim(Square, [N,N]),
Square[1..N,1..N] :: 1..Max,
Rows is Square[1..N,1..N],
flatten(Rows, Vars),
alldifferent(Vars),
Coding Magic square pb
%constraints on rows
(
for(I,1,N),
foreach(U,UpDiag),
foreach(D,DownDiag),
param(N,Square,Sum)
do
Magicsum #= sum(Square[I,1..N]),
Magicsum #= sum(Square[1..N,I]),
U is Square[I,I],
D is Square[I,N+1-I]
),
%constraints on diagonals
Magicsum #= sum(UpDiag),
Magicsum #= sum(DownDiag),
Coding Magic square pb
%remove symmetry
Square[1,1] #< Square[1,N],
Square[1,1] #< Square[N,N],
Square[1,1] #< Square[N,1],
Square[1,N] #< Square[N,1],
%search
labeling(Vars),
print(Square).
Unfortunately, ECLiPSe ran for a long time without
providing any answer. ECLiPSe had also the same behavior
for a similar code from ECLiPSe website.
A Tiny Transportation problem

How much should be shipped from several sources to
several destinations
Supply
cpty
Source
Demand
Qty Shipped Destination Qty
a1
1
a2
2 x21 x22
x
: 2n
m
:
am
x1n x12
x11
1
b1
2
:
n
b2
:
bn
A Tiny Transportation problem

3 plants with known capacities, 4 clients with known
demands and transport costs per unit between them.
500
300
400
1
2
3
Find qty shipped?
1
200
2
400
3
300
4
100
Modeling TTP

TTP can be modeled by a CSP with optimization as
follows:



Variables: A1,A2,A3,B1,B2,B3,C1, C2, C3,D1,D2,D3
Domain: 0.0..Inf
constraints:

Demand constraints:
A1+A2+A3=200; B1+B2+B3=400; C1+C2+C3=300; D1+D2+D3=100;

Capacity constraints:
A1+B1+C1+D1500; A2+B2+C2+D2300; A3+B3+C3+D3400;

Minimization of the objective function:
10*A1 + 7*A2 + 11*A3 + 8*B1 + 5*B2 + 10*B3 +
5*C1 + 5*C2 + 8*C3 + 9*D1 + 3*D2 + 7*D3
Coding TTP

OPL Studio - Cplex Solver
var float+ productA[Range];var float+ productB[Range];
var float+ productC[Range]; var float+ productD[Range];
minimize
10*productA[1] + 7*productA[2] + 11*productA[3] +
8*productB[1] + 5*productB[2] + 10*productB[3] +
5*productC[1] + 5*productC[2] + 8*productC[3] +
9*productD[1] + 3*productD[2] + 7*productD[3]
subject to {
productA[1] + productA[2] + productA[3] = 200;
productB[1] + productB[2] + productB[3] = 400;
productC[1] + productC[2] + productC[3] = 300;
productD[1] + productD[2] + productD[3] = 100;
productA[1] + productB[1] + productC[1] + productD[1]<=500
productA[2] + productB[2] + productC[2]+productC[2] <= 300;
productA[3] + productB[3] + productC[3] + productD[3] <= 400;
};
Coding TTP

OPL Studio - Cplex Solver: Solution found
Optimal Solution with Objective Value: 6200.0000
productA[1] = 100.0000
productA[2] = 0.0000
productA[3] = 100.0000
productB[1] = 100.0000
productB[2] = 300.0000
productB[3] = 0.0000
productC[1] = 300.0000
productC[2] = 0.0000
productC[3] = 0.0000
productD[1] = 0.0000
productD[2] = 100.0000
productD[3] = 0.0000
Coding TTP

ECLiPSe
:- lib(eplex_cplex).
main1(Cost, Vars) :Vars = [A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3],
Vars :: 0.0..inf,
A1 + A2 + A3 $= 200, B1 + B2 + B3 $= 400,
C1 + C2 + C3 $= 300, D1 + D2 + D3 $= 100,
A1 + B1 + C1 + D1 $=< 500, A2 + B2 + C2 + D2 $=< 300,
A3 + B3 + C3 + D3 $=< 400,
optimize(min(10*A1 + 7*A2 + 11*A3 + 8*B1 + 5*B2 + 10*B3 +
5*C1 + 5*C2 + 8*C3 +9*D1 + 3*D2 + 7*D3), Cost).
It didn’t run!
calling an undefined procedure cplex_prob_init(…)
Zebra puzzle




Zebra is a well known puzzle in which five men with different
nationalities live in the first five house of a street. They
practice five distinct professions, and each of them has a
favorite animal and a favorite drink, all of them different. The
five houses are painted in different colors. The puzzle is to
find who owns Zebra.
Zebra puzzle has different statements. We’ll handle two.
This kind of Puzzle is usually treated as an instance of the
class of tabular constraint satisfaction problems in which we
express the problem using tables.
In this presentation we show how to solve it using CSP
languages: OPL studio and ECLiPSe.
Modeling Zebra puzzle

In crucial step in Modeling Zebra puzzle is finding the decision
variables. In our case, we consider:
 variables: persons, colors, pets, drinks, tobaccos.
 Domain: 1..5 (representing the five houses)
 Constraints:
Expressed directly from the statement of the puzzle.
Coding Zebra puzzle

OPL Studio - ILog Solver
enum people {Englishman, Spaniard, Ukrainian, Norwegian, Japanese};
enum drinks {coffee, tea, milk, orange, water};
enum pets {dog, fox, zebra, horse, snails};
enum tabacco {winston, kools, chesterfields, luckyStrike, parliaments };
enum colors {ivory, yellow, red , green, blue};
range houses 1..5;//{first, second, third, fourth, fifth};
var
var
var
var
var
houses
houses
houses
houses
houses
hseClr[colors];
hsePple[people];
hsePts[pets];
hseDrks[drinks];
hseTaba[tabacco];
Coding Zebra puzzle
solve{
hsePple[Englishman] = hseClr[red]; hsePple[Spaniard]=hsePts[dog];
hseDrks[coffee] = hseClr[green]; hsePple[Ukrainian] = hseDrks[tea];
hseClr[green] = hseClr[ivory]+1; hseTaba[winston] = hsePts[snails];
hseTaba[kools]=hseClr[yellow]; hseDrks[milk]=3;
hsePple[Norwegian] = 1;
hseTaba[chesterfields]=hsePts[fox]+1 \/ hseTaba[chesterfields]=hsePts[fox]-1;
hseTaba[kools] = hsePts[horse]+1 \/ hseTaba[kools] = hsePts[horse]-1 ;
hseTaba[ luckyStrike] = hseDrks[orange];
hsePple[ Japanese] = hseTaba[parliaments];
hsePple[Norwegian] = hseClr[blue]+1 \/ hsePple[Norwegian] = hseClr[blue]-1;
//slient constraints-Global constraint alldifferent
alldifferent(hsePple);//forall(ordered i,j in people) hsePple[i] <> hsePple[j];
alldifferent(hsePts);//forall(ordered i,j in pets) hsePts[i] <> hsePts[j];
alldifferent(hseClr);//forall(ordered i,j in colors) hseClr[i] <> hseClr[j];
alldifferent(hseDrks);//forall(ordered i,j in drinks) hseDrks[i] <> hseDrks[j];
alldifferent(hseTaba);//forall(ordered i,j in tabacco) hseTaba[i] <> hseTaba[j];
};
Coding Zebra puzzle
% The Englishman lives in a red house.
% The Spaniard owns a dog.
% The Japanese is a painter.
% The Italian drinks tea.
% The Norwegian lives in the first house on the left.
% The owner of the green house drinks coffee.
% The green house is on the right of the white one.
% The sculptor breeds snails.
% The diplomat lives in the yellow house.
% Milk is drunk in the middle house.
% The Norwegian's house is next to the blue one.
% The violinist drinks fruit juice.
% The fox is in a house next to that of the doctor.
% The horse is in a house next to that of the diplomat.
%
% Who owns a Zebra, and who drinks water?
%
Coding Zebra puzzle
:- lib(fd).
zebra :% we use 5 lists of 5 variables each
Nat = [English, Spaniard, Japanese, Italian, Norwegian],
Color = [Red, Green, White, Yellow, Blue],
Profession = [Painter, Sculptor, Diplomat, Violinist, Doctor],
Pet = [Dog, Snails, Fox, Horse, Zebra],
Drink = [Tea, Coffee, Milk, Juice, Water],
% domains: all the variables range over house numbers 1 to 5
Nat :: 1..5,
Color :: 1..5,
Profession :: 1..5,
Pet :: 1..5,
Drink :: 1..5,
Coding Zebra puzzle
% the values in each list are exclusive
alldifferent(Nat),
alldifferent(Color),
alldifferent(Profession),
alldifferent(Pet),
alldifferent(Drink),
% and here follow the actual constraints
English = Red, Spaniard = Dog,
Japanese = Painter, Italian = Tea,
Norwegian = 1, Green = Coffee,
Green #= White + 1, Sculptor = Snails,
Diplomat = Yellow, Milk = 3,
Dist1 #= Norwegian - Blue, Dist1 :: [-1, 1],
Violinist = Juice,
Dist2 #= Fox - Doctor, Dist2 :: [-1, 1],
Dist3 #= Horse - Diplomat, Dist3 :: [-1, 1],
Coding Zebra puzzle
% put all the variables in a single list
flatten([Nat, Color, Profession, Pet, Drink], List),
% search: label all variables with values
labeling(List),
% print the answers: we need to do some decoding
NatNames = [English-english, Spaniard-spaniard, Japanese-japanese,
Italian-italian, Norwegian-norwegian],
memberchk(Zebra-ZebraNat, NatNames),
memberchk(Water-WaterNat, NatNames),
printf("The %w owns the zebra%n", [ZebraNat]),
printf("The %w drinks water%n", [WaterNat]).
Answer from ECLiPSe after 0.02s
The japanese owns the zebra
The norwegian drinks water
Uzbekian Puzzle
An uzbekian sales man met five traders who live in five
different cities. The five traders are:
 {Abdulhamid, Kurban,Ruza, Sharaf, Usman}
The five cities are :
 {Bukhara, Fergana, Kokand, Samarkand, Tashkent}

Find the order in which he visited the cities given the following
information:





He met Ruza before Sharaf after visiting Samarkand,
He reached Fergana after visiting Samarkand followed by other two
cities,
The third trader he met was Tashkent,
Immediately after his visit to Bukhara, he met Abdulhamid
He reached Kokand after visiting the city of Kurban followed by other
two cities;
Modeling Uzbekian Puzzle

The uzbekian puzzle can formulated within the CSP
framework as follows:
 Variables: order in which he visited each city and met
each trader
 Domain:1..5
 constraints:





He met Ruza before Sharaf after visiting Samarkand,
He reached Fergana after visiting Samarkand followed by other
two cities,
The third trader he met was Tashkent,
Immediately after his visit to Bukhara, he met Abdulhamid
He reached Kokand after visiting the city of Kurban followed by
other two cities;
Coding Uzbekian Puzzle in OPL
enum cities {Bukhara, Fergana, Kokand, Samarkand, Tashkent};
enum traders {Abdulhamid, Kurban, Ruza, Sharaf, Usman};
range order 1..5;
var order mtTrader[traders];
var order vstCty[cities];
Solution [1]
solve{
mtTrader[Ruza] < mtTrader[Sharaf];
mtTrader[Abdulhamid] = 2
mtTrader[Ruza] > vstCty[Samarkand];
mtTrader[Kurban] = 3
vstCty[Fergana] = vstCty[Samarkand] + 2;
mtTrader[Ruza] = 4
vstCty[Tashkent] = 3;
mtTrader[Sharaf] = 5
mtTrader[Abdulhamid] = vstCty[Bukhara] + 1;
mtTrader[Usman] = 1
vstCty[Kokand] = mtTrader[Kurban] + 2;
alldifferent(mtTrader);
vstCty[Bukhara] = 1
alldifferent(vstCty);
vstCty[Fergana] = 4
};
vstCty[Kokand] = 5
vstCty[Samarkand] = 2
vstCty[Tashkent] = 3
Knapsack problem


We have a knapsack with a fixed capacity and a number of
items. Each item has a weight and a value. The problem
consists of filling the knapsack without exceeding its
capacity, while maximizing the overall value of its contents.
Knapsack problem is an example of Mixed integer
programming.
Modeling Knapsack problem

A CSP model for knapsack problem is given by:




Variables: For each item, we associate a variable that gives the
quantity of such an item we can put in the knapsack.
Domain: 0..Capacity of the knapsack
Constraints:
 sum of the weights in the knapsack is less than the capacity
Objective function to maximize:
 sum of the values in the knapsack
Coding Knapsack Pb in OPL
range items 1..10;
int MaxCapacity= 1000;
int value[items] = [29,30,25,27, 28,21,23,19,18,17];
int weight[items] =[30,32,33,31,34,40,45,44,39,35];
var int take[items] in 0..MaxCapacity;
maximize
sum(i in items) value[i] * take[i]
subject to
sum(i in items) weight[i] * take[i] <= MaxCapacity;
integer programming (CPLEX MIP)
displaying solution ...
take[1] = 28 take[2] = 5 take[i] = 0, i=3..10
Graceful labeling problem


Given a tree T with m+1 nodes, and m edges, find the
possible labels in {0,1,…,m} for the nodes such that the
absolute value of the difference of the labels of the nodes
related to each edge are all different.
Formally, let f be a function from V ----> {0,…,m}, where V
is the set of the vertices of T.
 f is said to be a graceful labeling of T iff
 |f(vi)-f(vj)| are all different for any edge vivj in T.
Modeling Graceful labeling Pb

Graceful labeling problem can be formulated into the
following CSP:
 variables: labels to put on each node of T
 Domain: 0..m
 constraints:
 the absolute value of the difference between the
labels of any edge are all different
Coding Graceful labeling Pb

OPL program for graceful labeling problem
int nbreOfNodes = ...;
//set the range of the labels of the nodes
range nodes 0..nbreOfNodes-1; range indices 1..nbreOfNodes;
int adjacencyMatrix[indices, indices] = ...;
var nodes label[indices];
solve {
alldifferent(label);
forall(ordered i, j in indices){
if(adjacencyMatrix[i,j] <> 0) then
forall(ordered k, h in indices){
if(adjacencyMatrix[k,h] <> 0 & not(k=i & h=j)) then
abs(label[i]-label[j])<>abs(label[h]-label[k])
endif;
}
endif; };};
Coding Graceful labeling Pb

Data file
nbreOfNodes = 7;
adjacencyMatrix =
[ 1, 0, 1, 0,
[ 0, 1, 0, 1,
[ 0, 0, 1, 0,
[ 0, 0, 0, 1,
[ 0, 0, 1, 0,
[ 0, 0, 0, 0,

1
[[ 0,
0, 0,
0, 1,
1, 0,
0, 0,
0, 0,
0, 1,
Solution Example:
1, 0, 0, 0, 0, 0],
0 ],
0 ],
0 ],
0 ],
5
1 ],
0 ]];
label[1] = 0
label[2] = 6
label[3] = 1
label[4] = 3
label[5] = 4
label[6] = 5
label[7] = 2
2
3
4
6
7
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Examples for Constraint Programming