Minimum Set Cover

I would like to solve a minimum set cover problem of the following sort. All the lists contain only 1s and 0s. I say that a list A covers a list B if you can make B from A by ins

Solution 1:

I’ve written a small program to write an integer program to be solved by CPLEX or another MIP solver. Below it is a solution for n=12.

---

from collections import defaultdict
from itertools import product, combinations

def all_fill(source, num):
    output_len = (len(source) + num)
    for where in combinations(range(output_len), len(source)):
        poss = ([[0, 1]] * output_len)
        for (w, s) in zip(where, source):
            poss[w] = [s]
        for tup in product(*poss):
            (yield tup)

def variable_name(seq):
    return ('x' + ''.join((str(s) for s in seq)))
n = 12
shortn = ((2 * n) // 3)
x = (n // 3)
all_seqs = list(product([0, 1], repeat=shortn))
hit_sets = defaultdict(set)
for seq in all_seqs:
    for fill in all_fill(seq, x):
        hit_sets[fill].add(seq)
print('Minimize')
print(' + '.join((variable_name(seq) for seq in all_seqs)))
print('Subject To')
for (fill, seqs) in hit_sets.items():
    print(' + '.join((variable_name(seq) for seq in seqs)), '>=', 1)
print('Binary')
for seq in all_seqs:
    print(variable_name(seq))
print('End')

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MIP - Integer optimal solution:  Objective =  1.0000000000e+01
Solution time =    7.66 sec.  Iterations = 47411  Nodes = 337

CPLEX> Incumbent solution
Variable Name           Solution Value
x00000000                     1.000000
x00000111                     1.000000
x00011110                     1.000000
x00111011                     1.000000
x10110001                     1.000000
x11000100                     1.000000
x11001110                     1.000000
x11100001                     1.000000
x11111000                     1.000000
x11111111                     1.000000
All other variables matching '*' are 0.
CPLEX>