examples/notebook/contrib/einav_puzzle.ipynb
Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
First, you must install ortools package in this colab.
%pip install ortools
A programming puzzle from Einav in Google CP Solver.
From 'A programming puzzle from Einav' http://gcanyon.wordpress.com/2009/10/28/a-programming-puzzle-from-einav/ ''' My friend Einav gave me this programming puzzle to work on. Given this array of positive and negative numbers: 33 30 -10 -6 18 7 -11 -23 6 ... -25 4 16 30 33 -23 -4 4 -23
You can flip the sign of entire rows and columns, as many of them as you like. The goal is to make all the rows and columns sum to positive numbers (or zero), and then to find the solution (there are more than one) that has the smallest overall sum. So for example, for this array: 33 30 -10 -16 19 9 -17 -12 -14 You could flip the sign for the bottom row to get this array: 33 30 -10 -16 19 9 17 12 14 Now all the rows and columns have positive sums, and the overall total is 108. But you could instead flip the second and third columns, and the second row, to get this array: 33 -30 10 16 19 9 -17 12 14 All the rows and columns still total positive, and the overall sum is just 66. So this solution is better (I don't know if it's the best) A pure brute force solution would have to try over 30 billion solutions. I wrote code to solve this in J. I'll post that separately. '''
Compare with the following models:
Note: einav_puzzle2.py is Laurent Perron version, which don't use as many decision variables as this version.
This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/
from ortools.constraint_solver import pywrapcp
def main():
# Create the solver.
solver = pywrapcp.Solver('Einav puzzle')
#
# data
#
# small problem
# rows = 3;
# cols = 3;
# data = [
# [ 33, 30, -10],
# [-16, 19, 9],
# [-17, -12, -14]
# ]
# Full problem
rows = 27
cols = 9
data = [[33, 30, 10, -6, 18, -7, -11, 23, -6],
[16, -19, 9, -26, -8, -19, -8, -21, -14],
[17, 12, -14, 31, -30, 13, -13, 19, 16],
[-6, -11, 1, 17, -12, -4, -7, 14, -21],
[18, -31, 34, -22, 17, -19, 20, 24, 6],
[33, -18, 17, -15, 31, -5, 3, 27, -3],
[-18, -20, -18, 31, 6, 4, -2, -12, 24],
[27, 14, 4, -29, -3, 5, -29, 8, -12],
[-15, -7, -23, 23, -9, -8, 6, 8, -12],
[33, -23, -19, -4, -8, -7, 11, -12, 31],
[-20, 19, -15, -30, 11, 32, 7, 14, -5],
[-23, 18, -32, -2, -31, -7, 8, 24, 16],
[32, -4, -10, -14, -6, -1, 0, 23, 23],
[25, 0, -23, 22, 12, 28, -27, 15, 4],
[-30, -13, -16, -3, -3, -32, -3, 27, -31],
[22, 1, 26, 4, -2, -13, 26, 17, 14],
[-9, -18, 3, -20, -27, -32, -11, 27, 13],
[-17, 33, -7, 19, -32, 13, -31, -2, -24],
[-31, 27, -31, -29, 15, 2, 29, -15, 33],
[-18, -23, 15, 28, 0, 30, -4, 12, -32],
[-3, 34, 27, -25, -18, 26, 1, 34, 26],
[-21, -31, -10, -13, -30, -17, -12, -26, 31],
[23, -31, -19, 21, -17, -10, 2, -23, 23],
[-3, 6, 0, -3, -32, 0, -10, -25, 14],
[-19, 9, 14, -27, 20, 15, -5, -27, 18],
[11, -6, 24, 7, -17, 26, 20, -31, -25],
[-25, 4, -16, 30, 33, 23, -4, -4, 23]]
#
# variables
#
x = {}
for i in range(rows):
for j in range(cols):
x[i, j] = solver.IntVar(-100, 100, 'x[%i,%i]' % (i, j))
x_flat = [x[i, j] for i in range(rows) for j in range(cols)]
row_sums = [solver.IntVar(0, 300, 'row_sums(%i)' % i) for i in range(rows)]
col_sums = [solver.IntVar(0, 300, 'col_sums(%i)' % j) for j in range(cols)]
row_signs = [solver.IntVar([-1, 1], 'row_signs(%i)' % i) for i in range(rows)]
col_signs = [solver.IntVar([-1, 1], 'col_signs(%i)' % j) for j in range(cols)]
# total sum: to be minimized
total_sum = solver.IntVar(0, 1000, 'total_sum')
#
# constraints
#
for i in range(rows):
for j in range(cols):
solver.Add(x[i, j] == data[i][j] * row_signs[i] * col_signs[j])
total_sum_a = [
data[i][j] * row_signs[i] * col_signs[j]
for i in range(rows)
for j in range(cols)
]
solver.Add(total_sum == solver.Sum(total_sum_a))
# row sums
for i in range(rows):
s = [row_signs[i] * col_signs[j] * data[i][j] for j in range(cols)]
solver.Add(row_sums[i] == solver.Sum(s))
# column sums
for j in range(cols):
s = [row_signs[i] * col_signs[j] * data[i][j] for i in range(rows)]
solver.Add(col_sums[j] == solver.Sum(s))
# objective
objective = solver.Minimize(total_sum, 1)
#
# search and result
#
# Note: The order of the variables makes a big difference.
# If row_signs are before col_sign it is much slower.
db = solver.Phase(col_signs + row_signs, solver.CHOOSE_MIN_SIZE_LOWEST_MIN,
solver.ASSIGN_MAX_VALUE)
solver.NewSearch(db, [objective])
num_solutions = 0
while solver.NextSolution():
num_solutions += 1
print('total_sum:', total_sum.Value())
print('row_sums:', [row_sums[i].Value() for i in range(rows)])
print('col_sums:', [col_sums[j].Value() for j in range(cols)])
print('row_signs:', [row_signs[i].Value() for i in range(rows)])
print('col_signs:', [col_signs[j].Value() for j in range(cols)])
print('x:')
for i in range(rows):
for j in range(cols):
print('%3i' % x[i, j].Value(), end=' ')
print()
print()
solver.EndSearch()
print()
print('num_solutions:', num_solutions)
print('failures:', solver.Failures())
print('branches:', solver.Branches())
print('WallTime:', solver.WallTime(), 'ms')
main()