import numpy as np

def solution(entrances, exits, path):
    p = np.array(path)
    minimum_width = -1
    all_rooms = set(range(len(p)))
    for rooms, others in generate_rooms(all_rooms, set(entrances), set(exits), p):
        width = calculate_cut_width(rooms, others, p)
        if width < minimum_width or minimum_width == -1:
            minimum_width = width
    return minimum_width

def generate_rooms(all_rooms, prev_rooms, exits, path):
    yield prev_rooms, all_rooms - set(prev_rooms)
    for room, width in enumerate(sum(path[list(prev_rooms)])):
        if room not in prev_rooms and room not in exits and width != 0:
            next_rooms = prev_rooms | set([room])
            for result in generate_rooms(all_rooms, next_rooms, exits, path):
                yield result

def calculate_cut_width(rooms, others, p):
    return np.sum(p[list(rooms)][:, list(others)])

tests = [
        [[[0], [3], [[0, 7, 0, 0], [0, 0, 6, 0], [0, 0, 0, 8], [9, 0, 0, 0]]], 6],
        [[[0, 1], [4, 5], [[0, 0, 4, 6, 0, 0], [0, 0, 5, 2, 0, 0], [0, 0, 0, 0, 4, 4], [0, 0, 0, 0, 6, 6], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]]], 16],
        ]

for test in tests:
    result = solution(*test[0])
    print(test[0], result == test[1], result, test[1])