Generate an m*n matrix with 1 and 0 (No identity)

Question

I need to generate an m*n matrix with 1 and 0 where the sum of each row is 1 and each column is greater than or equal to 1. I'm trying this code but I don't know how to limit my condition on the columns, since sometimes I get zero columns and I need them all to have at least one 1. How could I limit the sum in the columns to be at least 1?

pt = []

sumr = 0
sumc = []

for i in range(n):
    
    quantity = random.choices([1])
    row = np.zeros(p, dtype=int)
    idx = np.random.choice(range(p), quantity, replace=False)
    row[idx] = 1
    pt.append(row)
    
for j in range(len(pt[0])):
    sumc.append(sum([row[j] for row in pt]))
    print(sumc)

Answer 1

I would just create a matrix with ones in random places, and then check if there are still zero columns once we near the end so we don't leave any empty:

import numpy as np
import random

def make_mat(m, n):
    if (n > m):
        print("Invalid matrix dimensions.")
        return
    
    mat = np.zeros((m,n))
    zero_cols = [True]*n   # keep track of zero columns
    
    for row in range(m):
        # if there are N zero cols and N rows remaining, place 1 at randomly selected zero column place
        if len(np.where(zero_cols)[0]) == (m - row):
            idx = random.choice(np.where(zero_cols)[0])
        # otherwise, just select a random index
        else:
            idx = random.randint(0, n-1)
        mat[row][idx] = 1
        zero_cols[idx] = False
    
    return mat

print(make_mat(10, 8))

However, this can be simplified if we make use of an identity matrix:

import numpy as np

def make_mat_id(m, n):
    if (n > m):
        print("Invalid matrix dimensions.")
        return
    
    id_mat = np.identity(n)
    # add rows with one in random place
    mat = np.concatenate((id_mat, [np.where(np.arange(n) == np.random.choice(np.arange(n)), 1, 0) for _ in range(m-n)]))
    # make sure we no longer have the identity matrix
    np.random.shuffle(mat)
    return mat

print(make_mat_id(10, 8)

Answer 2

If I understood correctly what you're trying to do, you could define a simple optimization problem and solve it to find an m*n matrix that fits to your constraints.

To do so, first you will have to install an optimization package. I recommend using a package like PuLP. To install PuLP, run the following command:

pip install pulp

Here's an example on how you can create your matrix, by defining and solving an optimization problem with PuLP:

import pandas as pd
import pulp


M = 6    # Number of rows in the Matrix
N = 5    # Number of columns in the Matrix
# IMPORTANT: `M` needs to be greater than or equal to `N`,
#            otherwise it's impossible to generate a matrix,
#            given your problem set of constraints.

# Create the problem instance
prob = pulp.LpProblem("MxN_Matrix", sense=pulp.LpMaximize)

# Generate the matrix with binary variables
matrix = pd.DataFrame(
    [
        [
            pulp.LpVariable(f"i{row}{col}", cat=pulp.LpBinary, upBound=1, lowBound=0)
            for col in range(1, N + 1)
        ]
        for row in range(1, M + 1)
    ]
)

# Set the constraints to the problem

# Constraint 1: Each row must have exactly 1 element equal to 1
for row_sum in matrix.sum(axis=1):
    prob += row_sum == 1

# Constraint 2: Each column must have at least 1 element equal to 1
for col_sum in matrix.sum(axis=0):
    prob += col_sum >= 1

# Set an arbitrary objective function
prob.setObjective(matrix.sum(axis=1).sum())

# Solve the problem
status = prob.solve()

# Print the status of the solution
print(pulp.LpStatus[status])

# Retrieve the solution
result = matrix.applymap(lambda value: value.varValue).astype(int)
print(result)
# Prints:
#
#  0  1  0  0  0
#  0  0  1  0  0
#  0  0  0  0  1
#  0  1  0  0  0
#  0  0  0  1  0
#  1  0  0  0  0

We can check that both your constraints are satisfied, by calculating the sum of each row, and column:

Sum of each row:

result.sum(axis=1)
# Returns:
#
# 0    1
# 1    1
# 2    1
# 3    1
# 4    1
# 5    1
# dtype: int64

Sum of each column:

result.sum()
# Returns:
#
# 0    1
# 1    2
# 2    1
# 3    1
# 4    1
# dtype: int64

NOTE

It's important to note that in order to create a matrix that satisfies the constraints of your problem, your matrix must have a number of rows greater than or equal to the number of columns. Since each column must have at least 1 element equal to 1 and each row needs to have exactly 1 element equal to 1, it's impossible for example to generate a matrix with 2 rows and 5 columns, since only 2 out of the 5 columns can contain an element that is not equal to zero.