Modeling Constrained Combinatorial Problems in Python

# PyCSP3 v2.0 (December 15, 2021)

This is Version 2.0 of PyCSP3, a library in Python 3 (version 3.6 or later) for modeling combinatorial constrained problems; see www.pycsp.org. PyCSP3 is inspired from both JvCSP3 (a Java-based API) and Numberjack. With PyCSP3, it is possible to generate instances of:

1. CSPs (Constraint Satisfaction Problems)
2. COPs (Constraint Optimization Problems)

in format XCSP3; see www.xcsp.org. Currently, PyCSP3 is targeted to XCSP3-core, which allows us to use integer variables (with finite domains) and popular constraints.

Note that:

• a dedicated website with more than 60 Jupyter notebooks is available
• a well-documented guide is available
• PyCSP3 is available as a PyPi package

At this stage, one can run two embedded solvers:

• the constraint solver ACE (AbsCon Essence), with the option -solve or the option -solver=ace
• the constraint solver Choco, with the option -solver=choco

Information about how piloting these embedded solvers can be found in this document.

Of course, it is possible to launch on generated XCSP3 instances (files) any solver that recognizes the XCSP3 format. It is, for example, immediate to run ACE or Choco on XCSP3 instances (files) as the respective executables (jar files) are present in directories pycsp3/solvers/ace and pycsp3/solvers/choco. For example, for running ACE on the XCSP3 instance 'zebra.xml', just execute:

java -jar ACE-YY-MM.jar zebra.xml


while replacing YY and MM with the current values that are present in the name of the jar file.

Note that it is also possible to pilot solvers with Python; see PyCSP3 Solving Process.

# 1) Installation from PyPi

This is the easiest way of installing PySCP3.

Note that you need first Python 3 (version 3.6, or later) to be installed. You can do it, for example, from python.org

## Installing PyCSP3 (Linux)

Check if 'pip3' is installed. If it is not the case, execute:

sudo apt install python3-pip


Then, install PyCSP3 with the command 'pip3':

sudo pip3 install pycsp3


For using the -solve or -solver options, you need to have Java (at least, version 11) installed:

sudo apt-get install openjdk-11-jdk


## Installing PyCSP3 (Mac OS)

If Python 3 is installed on your system, the command 'pip3' should already be present.

Install PyCSP3 with the command 'pip3':

sudo pip3 install pycsp3


For using the -solve or -solver options, you need to have Java (at least, version 11) installed.

## Installing PyCSP3 (Windows)

You may need to upgrade 'pip'. Open the console and type:

python -m pip install --upgrade pip


Then, for installing pycsp3, type:

python -m pip install pycsp3


For using the -solve or -solver options, you need to install (at least) Java version 11:

And add in the PATH the java command, for example, temporally, with the command:

set path=%path%;C:/Program Files/Java/jdk-14.0.1/bin/


You can check the java command by typing in your console:

java --version


## Updating the Version of PyCSP3 (for PyPi)

For updating your version of PyCSP3, simply execute:

For linux/Mac:

sudo pip3 install --upgrade pycsp3


For Windows:

python -m pip install --upgrade pycsp3


## Copying a Pool of Models

PyCSP3 is accompanied by more than 100 models. To get them in a subdirectory problems of your current directory, execute:

python3 -m pycsp3 (For linux/Mac)
python -m pycsp3 (For Windows)


And you can test the compilation of one of the models, for example:

python3 problems/csp/single/Zebra.py (For Linux/Mac)
python problems\csp\single\Zebra.py (For Windows)


# 2) Installation (alternative) by Cloning from GitHub

An alternative to PyPi is to clone the code from GitHub. Here is an illustration for MAC OS. We assume that Python 3 is installed (otherwise, type port install python38 for example), and consequently 'pip3' is also installed. In a console, type:

git clone https://github.com/xcsp3team/pycsp3.git
pip3 install lxml


You may need to update the environment variable 'PYTHONPATH', by typing for example:

export PYTHONPATH=$PYTHONPATH:.  # 3) Compilation and Examples We succinctly introduce a few PyCSP3 models, showing how to compile them with different options. However, note that many illustrations are available on www.pycsp.org, notably many models with Jupyter notebooks. First, we give some general information about compilation. ## Compiling PyCSP3 Models For generating an XCSP3 instance from a PyCSP3 model, you have to execute: python3 <file> [options]  with: • <file>: a Python file to be executed, describing a model in PyCSP3 • [options]: possible options to be used when compiling Among the options, we find: • -data=<data_value>: allows us to specify the data to be used by the model. It can be: • elementary: -data=5 • a simple list: -data=[9,0,0,3,9] • a JSON file: -data=Bibd-3-4-6.json Data can then be directly used in the PyCSP3 model by means of a predefined variable data. • -dataparser=<file>: a Python file for reading/parsing data given under any arbitrary form (e.g., by a text file). See Example Nonogram below, for an illustration. • -dataexport: exports (saves) the data in JSON format. See Example Nonogram below, for an illustration. • -variant=<variant_name>: the name of a variant, to be used with function variant(). See Example AllInterval below, for an illustration. • -solve: attempts to solve the instance with the embedded solver 'Ace'. It requires that Java version 8 (at least) is installed. • -solver=<solver_name>: attempts to solve the instance with the solver whose name is given. Currently, it can be 'ace' or 'choco'. Important: it requires that Java version 8 (at least) is installed. Information about how piloting these embedded solvers can be found in this document. • -output=<file_name>: sets the filename of the generated XCSP3 instance (think about the extension .xml) By default, a file containing the XCSP3 instance is generated, unless you use the option: • -display: displays the XCSP3 instance in the system standard output, instead of generating an XCSP3 file ## Example 1: in console mode Our first example shows how you can build basic models in console mode. In this example, we just post two variable and two simple binary constraints. $ python3
Python 3.5.2
>>> from pycsp3 import *
>>> x = Var(range(10))
>>> y = Var(range(10))
>>> satisfy(
x < y,
x + y > 15
)
>>> compile()


Note that to get an XCSP3 file, we call compile().

## Example 2: Send+More=Money

This example shows how you can define a model when no data is required from the user. This is the classical crypto-arithmetic puzzle 'Send+More=Money'.

#### File SendMore.py

from pycsp3 import *

# letters[i] is the digit of the ith letter involved in the equation
s, e, n, d, m, o, r, y = letters = VarArray(size=8, dom=range(10))

satisfy(
# letters are given different values
AllDifferent(letters),

[s > 0, m > 0],

# respecting the mathematical equation
[s, e, n, d] * [1000, 100, 10, 1]
+ [m, o, r, e] * [1000, 100, 10, 1]
== [m, o, n, e, y] * [10000, 1000, 100, 10, 1]
)


To generate the XCSP3 instance (file), the command is:

python3 SendMore.py


To generate and solve (with ACE) the XCSP3 instance, the command is:

python3 SendMore.py -solve


To generate and solve with Choco the XCSP3 instance, the command is:

python3 SendMore.py -solver=choco


## Example 3: All-Interval Series

This example shows how you can simply specify an integer (as unique data) for a model. For our illustration, we consider the problem All-Interval Series.

A classical model is:

#### File AllInterval.py (version 1)

from pycsp3 import *

n = data

# x[i] is the ith note of the series
x = VarArray(size=n, dom=range(n))

satisfy(
# notes must occur once, and so form a permutation
AllDifferent(x),

# intervals between neighbouring notes must form a permutation
AllDifferent(abs(x[i] - x[i + 1]) for i in range(n - 1)),

# tag(symmetry-breaking)
x[0] < x[n - 1]
)


Note the presence of a tag symmetry-breaking that will be directly integrated into the XCSP3 file generated by the following command:

python3 AllInterval.py -data=5


Suppose that you would prefer to declare a second array of variables for representing successive distances. This would give:

#### File AllInterval.py (version 2)

from pycsp3 import *

n = data

# x[i] is the ith note of the series
x = VarArray(size=n, dom=range(n))

# y[i] is the distance between x[i] and x[i+1]
y = VarArray(size=n - 1, dom=range(1, n))

satisfy(
# notes must occur once, and so form a permutation
AllDifferent(x),

# intervals between neighbouring notes must form a permutation
AllDifferent(y),

# computing distances
[y[i] == abs(x[i] - x[i + 1]) for i in range(n - 1)],

# tag(symmetry-breaking)
[x[0] < x[n - 1], y[0] < y[1]]
)


However, sometimes, it may be relevant to combine different variants of a model in the same file. In our example, this would give:

#### File AllInterval.py (version 3)

from pycsp3 import *

n = data

# x[i] is the ith note of the series
x = VarArray(size=n, dom=range(n))

if not variant():

satisfy(
# notes must occur once, and so form a permutation
AllDifferent(x),

# intervals between neighbouring notes must form a permutation
AllDifferent(abs(x[i] - x[i + 1]) for i in range(n - 1)),

# tag(symmetry-breaking)
x[0] < x[n - 1]
)

elif variant("aux"):

# y[i] is the distance between x[i] and x[i+1]
y = VarArray(size=n - 1, dom=range(1, n))

satisfy(
# notes must occur once, and so form a permutation
AllDifferent(x),

# intervals between neighbouring notes must form a permutation
AllDifferent(y),

# computing distances
[y[i] == abs(x[i] - x[i + 1]) for i in range(n - 1)],

# tag(symmetry-breaking)
[x[0] < x[n - 1], y[0] < y[1]]
)


For compiling the main model (variant), the command is:

python3 AllInterval.py -data=5


For compiling the second model variant, using the option -variant, the command is:

python3 AllInterval.py -data=5 -variant=aux


To generate and solve (with ACE) the instance of order 10 and variant 'aux', the command is:

python3 AllInterval.py -data=10 -variant=aux -solve


## Example 4: BIBD

This example shows how you can specify a list of integers to be used as data for a model. For our illustration, we consider the problem BIBD. We need five integers v, b, r, k, l for specifying a unique instance (possibly, b and r can be set to 0, so that these values are automatically computed according to a template for this problem). The model is:

#### File Bibd.py

from pycsp3 import *

v, b, r, k, l = data
b = (l * v * (v - 1)) // (k * (k - 1)) if b == 0 else b
r = (l * (v - 1)) // (k - 1) if r == 0 else r

# x[i][j] is the value of the matrix at row i and column j
x = VarArray(size=[v, b], dom={0, 1})

satisfy(
# constraints on rows
[Sum(row) == r for row in x],

# constraints on columns
[Sum(col) == k for col in columns(x)],

# scalar constraints with respect to lambda
[row1 * row2 == l for (row1, row2) in combinations(x, 2)]
)


To generate an XCSP3 instance (file), we can for example execute a command like:

python3 Bibd.py -data=[9,0,0,3,9]


With some command interpreters (shells), you may have to escape the characters '[' and ']', which gives:

python3 Bibd.py -data=$9,0,0,3,9$


## Example 5: Rack Configuration

This example shows how you can specify a JSON file to be used as data for a model. For our illustration, we consider the problem Rack Configuration. The data (for a specific instance) are then initially given in a JSON file, as for example:

#### File Rack_r2.json

{
"nRacks": 10,
"models": [[150,8,150],[200,16,200]],
"cardTypes": [[20,20],[40,8],[50,4],[75,2]]
}


In the following model, we directly unpack the components of the variable data (because it is automatically given under the form of a named tuple) whose fields are exactly those of the main object in the JSON file.

#### File Rack.py

from pycsp3 import *

nRacks, models, cardTypes = data
models.append([0, 0, 0])  # we add first a dummy model (0,0,0)
powers, sizes, costs = zip(*models)
cardPowers, cardDemands = zip(*cardTypes)
nModels, nTypes = len(models), len(cardTypes)

table = {(i, powers[i], sizes[i], costs[i]) for i in range(nModels)}

# m[i] is the model used for the ith rack
m = VarArray(size=nRacks, dom=range(nModels))

# p[i] is the power of the model used for the ith rack
p = VarArray(size=nRacks, dom=powers)

# s[i] is the size (number of connectors) of the model used for the ith rack
s = VarArray(size=nRacks, dom=sizes)

# c[i] is the cost (price) of the model used for the ith rack
c = VarArray(size=nRacks, dom=costs)

# nc[i][j] is the number of cards of type j put in the ith rack
nc = VarArray(size=[nRacks, nTypes], dom=lambda i, j: range(min(max(sizes), cardDemands[j]) + 1))

satisfy(
# linking rack models with powers, sizes and costs
[(m[i], p[i], s[i], c[i]) in table for i in range(nRacks)],

# connector-capacity constraints
[Sum(nc[i]) <= s[i] for i in range(nRacks)],

# power-capacity constraints
[nc[i] * cardPowers <= p[i] for i in range(nRacks)],

# demand constraints
[Sum(nc[:, j]) == cardDemands[j] for j in range(nTypes)],

# tag(symmetry-breaking)
[Decreasing(m), imply(m[0] == m[1], nc[0][0] >= nc[1][0])]
)

minimize(
# minimizing the total cost being paid for all racks
Sum(c)
)


To generate an XCSP3 instance (file), we execute the command:

python3 Rack.py -data=Rack_r2.json


One might want to have the data in the JSON file with another structure, as for example:

#### File Rack_r2b.json

{
"nRacks": 10,
"rackModels": [
{"power":150,"nConnectors":8,"price":150},
{"power":200,"nConnectors":16,"price":200}
],
"cardTypes": [
{"power":20,"demand":20},
{"power":40,"demand":8},
{"power":50,"demand":4},
{"power":75,"demand":2}
]
}


We only need to modify one line from the previous model:

#### File Rack2.py

models.append(models[0].__class__(0, 0, 0))  # we add first a dummy model (0,0,0) ; we get the class of the used named tuples to build a new one


To generate an XCSP3 instance (file), we execute the command:

python3 Rack2.py -data=Rack_r2b.json


## Example 6: Nonogram

This example shows how you can use an auxiliary Python file for parsing data that are not initially given under JSON format. For our illustration, we consider the problem Nonogram. The data (for a specific Nonogram puzzle) are initially given in a text file as follows:

1. a line stating the numbers of rows and columns,
2. then, for each row a line stating the number of blocks followed by the sizes of all these blocks (on the same line),
3. then, for each column a line stating the number of blocks followed by the sizes of all these blocks (on the same line).

Below, here is an example of such a text file.

#### File Nonogram_example.txt

24 24
0
1	5
2	3 3
2	1 2
2	2 1
2	1 1
2	3 3
3	1 5 1
3	1 1 1
3	2 1 1
3	1 1 2
3	3 1 3
3	1 3 1
3	1 1 1
3	2 1 2
3	1 1 1
1	5
3	1 1 1
3	1 1 1
3	1 1 1
3	5 1 1
2	1 2
3	2 2 4
2	4 9

0
0
0
1	1
1	2
1	2
2	6 1
3	3 1 3
3	1 1 4
4	2 1 1 7
5	1 1 1 1 1
3	1 12 1
5	1 1 1 1 1
4	2 1 1 7
4	1 1 4 1
4	2 1 2 2
2	8 3
2	1 1
2	1 2
1	4
1	3
1	2
1	1
0


First, we need to write a piece of code in Python for building a dictionary data that will be then used in our model (after having been automatically converted to a named tuple). We have first to import everything (*) from pycsp3.problems.data.parsing. We can then add any new arbitrary item to the dictionary data (which is initially empty). This is what we do below with two items whose keys are called rowPatterns and colPatterns. The values associated with these two keys are defined as two-dimensional arrays (lists) of integers, defining the sizes of blocks. The function next_int() can be called for reading the next integer in a text file, which will be specified on the command line (see later).

#### File Nonogram_Parser.py

from pycsp3.problems.data.parsing import *

nRows, nCols = next_int(), next_int()
data["rowPatterns"] = [[next_int() for _ in range(next_int())] for _ in range(nRows)]
data["colPatterns"] = [[next_int() for _ in range(next_int())] for _ in range(nRows)]


Then, we just write the model by getting data from the variable data. The model is totally independent of the way data were initially given (from a text file or a JSON file, for example). In the code below, note how an object Automaton is defined from a specified pattern (list of blocks). Also, for a regular constraint, we just write something like scope in automaton. Finally, x[:, j] denotes the jth column of x.

#### File Nonogram.py

from pycsp3 import *

rows, cols = data  # patterns for row and columns
nRows, nCols = len(rows), len(cols)

def automaton(pattern):
q = Automaton.q  # for building state names
transitions = []
if len(pattern) == 0:
n_states = 1
transitions.append((q(0), 0, q(0)))
else:
n_states = sum(pattern) + len(pattern)
num = 0
for i, size in enumerate(pattern):
transitions.append((q(num), 0, q(num)))
transitions.extend((q(num + j), 1, q(num + j + 1)) for j in range(size))
transitions.append((q(num + size), 0, q(num + size + (1 if i < len(pattern) - 1 else 0))))
num += size + 1
return Automaton(start=q(0), final=q(n_states - 1), transitions=transitions)

# x[i][j] is 1 iff the cell at row i and col j is colored in black
x = VarArray(size=[nRows, nCols], dom={0, 1})

satisfy(
[x[i] in automaton(rows[i]) for i in range(nRows)],

[x[:, j] in automaton(cols[j]) for j in range(nCols)]
)


To generate the XCSP3 instance (file), we just need to specify the name of the text file (option -data) and the name of the Python parser (option -dataparser).

python3 Nonogram.py -data=Nonogram_example.txt -dataparser=Nonogram_Parser.py


Maybe, you think that it would be simpler to have directly the data in JSON file. You can generate such a file with the option -dataexport. The command is as follows:

python3 Nonogram.py -data=Nonogram_example.txt -dataparser=Nonogram_Parser.py -dataexport


A file Nonogram_example.json is generated, whose content is:

{
"colPatterns":[[],[],[],[1],[2],[2],[6,1],[3,1,3],[1,1,4],[2,1,1,7],[1,1,1,1,1],[1,12,1],[1,1,1,1,1],[2,1,1,7],[1,1,4,1],[2,1,2,2],[8,3],[1,1],[1,2],[4],[3],[2],[1],[]],
"rowPatterns":[[],[5],[3,3],[1,2],[2,1],[1,1],[3,3],[1,5,1],[1,1,1],[2,1,1],[1,1,2],[3,1,3],[1,3,1],[1,1,1],[2,1,2],[1,1,1],[5],[1,1,1],[1,1,1],[1,1,1],[5,1,1],[1,2],[2,2,4],[4,9]]
}


With this new file, you can directly generate the XCSP3 file with:

python3 Nonogram.py -data=Nonogram_example.json


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