clustering_sat.py

# Copyright 2010-2018 Google LLC
# 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.
"""Cluster 40 cities in 4 equal groups to minimize sum of crossed distances."""

from __future__ import print_function
from __future__ import division

from ortools.sat.python import cp_model


distance_matrix = [
    [0, 10938, 4542, 2835, 29441, 2171, 1611, 9208, 9528, 11111, 16120, 22606, 22127, 20627, 21246, 23387, 16697, 33609, 26184, 24772, 22644, 20655, 30492, 23296, 32979, 18141, 19248, 17129, 17192, 15645, 12658, 11210, 12094, 13175, 18162, 4968, 12308, 10084, 13026, 15056],
    [10938, 0, 6422, 9742, 18988, 12974, 11216, 19715, 19004, 18271, 25070, 31971, 31632, 30571, 31578, 33841, 27315, 43964, 36944, 35689, 33569, 31481, 41360, 33760, 43631, 28730, 29976, 27803, 28076, 26408, 23504, 22025, 22000, 13197, 14936, 15146, 23246, 20956, 23963, 25994],
    [4542, 6422, 0, 3644, 25173, 6552, 5092, 13584, 13372, 13766, 19805, 26537, 26117, 24804, 25590, 27784, 21148, 37981, 30693, 29315, 27148, 25071, 34943, 27472, 37281, 22389, 23592, 21433, 21655, 20011, 17087, 15612, 15872, 11653, 15666, 8842, 16843, 14618, 17563, 19589],
    [2835, 9742, 3644, 0, 28681, 3851, 4341, 11660, 12294, 13912, 18893, 25283, 24777, 23173, 23636, 25696, 18950, 35927, 28233, 26543, 24127, 21864, 31765, 24018, 33904, 19005, 20295, 18105, 18551, 16763, 13958, 12459, 12296, 10370, 15331, 5430, 14044, 12135, 14771, 16743],
    [29441, 18988, 25173, 28681, 0, 31590, 29265, 37173, 35501, 32929, 40239, 47006, 46892, 46542, 48112, 50506, 44539, 60103, 54208, 53557, 51878, 50074, 59849, 52645, 62415, 47544, 48689, 46560, 46567, 45086, 42083, 40648, 40971, 29929, 28493, 34015, 41473, 38935, 42160, 44198],
    [2171, 12974, 6552, 3851, 31590, 0, 3046, 7856, 8864, 11330, 15411, 21597, 21065, 19382, 19791, 21845, 15099, 32076, 24425, 22848, 20600, 18537, 28396, 21125, 30825, 15975, 17101, 14971, 15104, 13503, 10544, 9080, 9983, 13435, 18755, 2947, 10344, 8306, 11069, 13078],
    [1611, 11216, 5092, 4341, 29265, 3046, 0, 8526, 8368, 9573, 14904, 21529, 21085, 19719, 20504, 22713, 16118, 32898, 25728, 24541, 22631, 20839, 30584, 23755, 33278, 18557, 19545, 17490, 17309, 15936, 12881, 11498, 12944, 14711, 19589, 5993, 12227, 9793, 12925, 14967],
    [9208, 19715, 13584, 11660, 37173, 7856, 8526, 0, 3248, 7855, 8245, 13843, 13272, 11526, 12038, 14201, 7599, 24411, 17259, 16387, 15050, 13999, 23134, 17899, 26460, 12894, 13251, 11680, 10455, 9997, 7194, 6574, 10678, 20959, 26458, 8180, 5255, 2615, 5730, 7552],
    [9528, 19004, 13372, 12294, 35501, 8864, 8368, 3248, 0, 4626, 6598, 13168, 12746, 11567, 12731, 15083, 9120, 25037, 18718, 18433, 17590, 16888, 25630, 20976, 29208, 16055, 16300, 14838, 13422, 13165, 10430, 9813, 13777, 22300, 27564, 10126, 8388, 5850, 8778, 10422],
    [11111, 18271, 13766, 13912, 32929, 11330, 9573, 7855, 4626, 0, 7318, 14185, 14005, 13655, 15438, 17849, 12839, 27179, 21947, 22230, 21814, 21366, 29754, 25555, 33535, 20674, 20872, 19457, 17961, 17787, 15048, 14372, 18115, 24280, 29101, 13400, 13008, 10467, 13375, 14935],
    [16120, 25070, 19805, 18893, 40239, 15411, 14904, 8245, 6598, 7318, 0, 6939, 6702, 6498, 8610, 10961, 7744, 19889, 15350, 16403, 16975, 17517, 24357, 22176, 28627, 18093, 17672, 16955, 14735, 15510, 13694, 13768, 18317, 28831, 34148, 16326, 11276, 9918, 11235, 11891],
    [22606, 31971, 26537, 25283, 47006, 21597, 21529, 13843, 13168, 14185, 6939, 0, 793, 3401, 5562, 6839, 8923, 13433, 11264, 13775, 15853, 17629, 21684, 22315, 26411, 19539, 18517, 18636, 16024, 17632, 16948, 17587, 22131, 34799, 40296, 21953, 14739, 14568, 14366, 14002],
    [22127, 31632, 26117, 24777, 46892, 21065, 21085, 13272, 12746, 14005, 6702, 793, 0, 2608, 4809, 6215, 8151, 13376, 10702, 13094, 15099, 16845, 21039, 21535, 25744, 18746, 17725, 17845, 15232, 16848, 16197, 16859, 21391, 34211, 39731, 21345, 14006, 13907, 13621, 13225],
    [20627, 30571, 24804, 23173, 46542, 19382, 19719, 11526, 11567, 13655, 6498, 3401, 2608, 0, 2556, 4611, 5630, 13586, 9157, 11005, 12681, 14285, 19044, 18996, 23644, 16138, 15126, 15240, 12625, 14264, 13736, 14482, 18958, 32292, 37879, 19391, 11621, 11803, 11188, 10671],
    [21246, 31578, 25590, 23636, 48112, 19791, 20504, 12038, 12731, 15438, 8610, 5562, 4809, 2556, 0, 2411, 4917, 12395, 6757, 8451, 10292, 12158, 16488, 16799, 21097, 14374, 13194, 13590, 10943, 12824, 12815, 13779, 18042, 32259, 37918, 19416, 10975, 11750, 10424, 9475],
    [23387, 33841, 27784, 25696, 50506, 21845, 22713, 14201, 15083, 17849, 10961, 6839, 6215, 4611, 2411, 0, 6760, 10232, 4567, 7010, 9607, 12003, 14846, 16408, 19592, 14727, 13336, 14109, 11507, 13611, 14104, 15222, 19237, 34013, 39703, 21271, 12528, 13657, 11907, 10633],
    [16697, 27315, 21148, 18950, 44539, 15099, 16118, 7599, 9120, 12839, 7744, 8923, 8151, 5630, 4917, 6760, 0, 16982, 9699, 9400, 9302, 9823, 16998, 14534, 21042, 10911, 10190, 9900, 7397, 8758, 8119, 8948, 13353, 27354, 33023, 14542, 6106, 6901, 5609, 5084],
    [33609, 43964, 37981, 35927, 60103, 32076, 32898, 24411, 25037, 27179, 19889, 13433, 13376, 13586, 12395, 10232, 16982, 0, 8843, 12398, 16193, 19383, 16423, 22583, 20997, 22888, 21194, 22640, 20334, 22636, 23801, 25065, 28675, 44048, 49756, 31426, 22528, 23862, 21861, 20315],
    [26184, 36944, 30693, 28233, 54208, 24425, 25728, 17259, 18718, 21947, 15350, 11264, 10702, 9157, 6757, 4567, 9699, 8843, 0, 3842, 7518, 10616, 10666, 14237, 15515, 14053, 12378, 13798, 11537, 13852, 15276, 16632, 19957, 35660, 41373, 23361, 14333, 16125, 13624, 11866],
    [24772, 35689, 29315, 26543, 53557, 22848, 24541, 16387, 18433, 22230, 16403, 13775, 13094, 11005, 8451, 7010, 9400, 12398, 3842, 0, 3795, 7014, 8053, 10398, 12657, 10633, 8889, 10569, 8646, 10938, 12906, 14366, 17106, 33171, 38858, 21390, 12507, 14748, 11781, 9802],
    [22644, 33569, 27148, 24127, 51878, 20600, 22631, 15050, 17590, 21814, 16975, 15853, 15099, 12681, 10292, 9607, 9302, 16193, 7518, 3795, 0, 3250, 8084, 6873, 11763, 6949, 5177, 7050, 5619, 7730, 10187, 11689, 13792, 30012, 35654, 18799, 10406, 12981, 9718, 7682],
    [20655, 31481, 25071, 21864, 50074, 18537, 20839, 13999, 16888, 21366, 17517, 17629, 16845, 14285, 12158, 12003, 9823, 19383, 10616, 7014, 3250, 0, 9901, 4746, 12531, 3737, 1961, 4036, 3588, 5109, 7996, 9459, 10846, 27094, 32690, 16451, 8887, 11624, 8304, 6471],
    [30492, 41360, 34943, 31765, 59849, 28396, 30584, 23134, 25630, 29754, 24357, 21684, 21039, 19044, 16488, 14846, 16998, 16423, 10666, 8053, 8084, 9901, 0, 9363, 4870, 13117, 11575, 13793, 13300, 15009, 17856, 19337, 20454, 36551, 42017, 26352, 18403, 21033, 17737, 15720],
    [23296, 33760, 27472, 24018, 52645, 21125, 23755, 17899, 20976, 25555, 22176, 22315, 21535, 18996, 16799, 16408, 14534, 22583, 14237, 10398, 6873, 4746, 9363, 0, 10020, 5211, 4685, 6348, 7636, 8010, 11074, 12315, 11926, 27537, 32880, 18634, 12644, 15358, 12200, 10674],
    [32979, 43631, 37281, 33904, 62415, 30825, 33278, 26460, 29208, 33535, 28627, 26411, 25744, 23644, 21097, 19592, 21042, 20997, 15515, 12657, 11763, 12531, 4870, 10020, 0, 14901, 13738, 15855, 16118, 17348, 20397, 21793, 21936, 37429, 42654, 28485, 21414, 24144, 20816, 18908],
    [18141, 28730, 22389, 19005, 47544, 15975, 18557, 12894, 16055, 20674, 18093, 19539, 18746, 16138, 14374, 14727, 10911, 22888, 14053, 10633, 6949, 3737, 13117, 5211, 14901, 0, 1777, 1217, 3528, 2896, 5892, 7104, 7338, 23517, 29068, 13583, 7667, 10304, 7330, 6204],
    [19248, 29976, 23592, 20295, 48689, 17101, 19545, 13251, 16300, 20872, 17672, 18517, 17725, 15126, 13194, 13336, 10190, 21194, 12378, 8889, 5177, 1961, 11575, 4685, 13738, 1777, 0, 2217, 2976, 3610, 6675, 8055, 8965, 25197, 30774, 14865, 8007, 10742, 7532, 6000],
    [17129, 27803, 21433, 18105, 46560, 14971, 17490, 11680, 14838, 19457, 16955, 18636, 17845, 15240, 13590, 14109, 9900, 22640, 13798, 10569, 7050, 4036, 13793, 6348, 15855, 1217, 2217, 0, 2647, 1686, 4726, 6000, 6810, 23060, 28665, 12674, 6450, 9094, 6117, 5066],
    [17192, 28076, 21655, 18551, 46567, 15104, 17309, 10455, 13422, 17961, 14735, 16024, 15232, 12625, 10943, 11507, 7397, 20334, 11537, 8646, 5619, 3588, 13300, 7636, 16118, 3528, 2976, 2647, 0, 2320, 4593, 6093, 8479, 24542, 30219, 13194, 5301, 8042, 4735, 3039],
    [15645, 26408, 20011, 16763, 45086, 13503, 15936, 9997, 13165, 17787, 15510, 17632, 16848, 14264, 12824, 13611, 8758, 22636, 13852, 10938, 7730, 5109, 15009, 8010, 17348, 2896, 3610, 1686, 2320, 0, 3086, 4444, 6169, 22301, 27963, 11344, 4780, 7408, 4488, 3721],
    [12658, 23504, 17087, 13958, 42083, 10544, 12881, 7194, 10430, 15048, 13694, 16948, 16197, 13736, 12815, 14104, 8119, 23801, 15276, 12906, 10187, 7996, 17856, 11074, 20397, 5892, 6675, 4726, 4593, 3086, 0, 1501, 5239, 20390, 26101, 8611, 2418, 4580, 2599, 3496],
    [11210, 22025, 15612, 12459, 40648, 9080, 11498, 6574, 9813, 14372, 13768, 17587, 16859, 14482, 13779, 15222, 8948, 25065, 16632, 14366, 11689, 9459, 19337, 12315, 21793, 7104, 8055, 6000, 6093, 4444, 1501, 0, 4608, 19032, 24747, 7110, 2860, 4072, 3355, 4772],
    [12094, 22000, 15872, 12296, 40971, 9983, 12944, 10678, 13777, 18115, 18317, 22131, 21391, 18958, 18042, 19237, 13353, 28675, 19957, 17106, 13792, 10846, 20454, 11926, 21936, 7338, 8965, 6810, 8479, 6169, 5239, 4608, 0, 16249, 21866, 7146, 7403, 8446, 7773, 8614],
    [13175, 13197, 11653, 10370, 29929, 13435, 14711, 20959, 22300, 24280, 28831, 34799, 34211, 32292, 32259, 34013, 27354, 44048, 35660, 33171, 30012, 27094, 36551, 27537, 37429, 23517, 25197, 23060, 24542, 22301, 20390, 19032, 16249, 0, 5714, 12901, 21524, 20543, 22186, 23805],
    [18162, 14936, 15666, 15331, 28493, 18755, 19589, 26458, 27564, 29101, 34148, 40296, 39731, 37879, 37918, 39703, 33023, 49756, 41373, 38858, 35654, 32690, 42017, 32880, 42654, 29068, 30774, 28665, 30219, 27963, 26101, 24747, 21866, 5714, 0, 18516, 27229, 26181, 27895, 29519],
    [4968, 15146, 8842, 5430, 34015, 2947, 5993, 8180, 10126, 13400, 16326, 21953, 21345, 19391, 19416, 21271, 14542, 31426, 23361, 21390, 18799, 16451, 26352, 18634, 28485, 13583, 14865, 12674, 13194, 11344, 8611, 7110, 7146, 12901, 18516, 0, 9029, 7668, 9742, 11614],
    [12308, 23246, 16843, 14044, 41473, 10344, 12227, 5255, 8388, 13008, 11276, 14739, 14006, 11621, 10975, 12528, 6106, 22528, 14333, 12507, 10406, 8887, 18403, 12644, 21414, 7667, 8007, 6450, 5301, 4780, 2418, 2860, 7403, 21524, 27229, 9029, 0, 2747, 726, 2749],
    [10084, 20956, 14618, 12135, 38935, 8306, 9793, 2615, 5850, 10467, 9918, 14568, 13907, 11803, 11750, 13657, 6901, 23862, 16125, 14748, 12981, 11624, 21033, 15358, 24144, 10304, 10742, 9094, 8042, 7408, 4580, 4072, 8446, 20543, 26181, 7668, 2747, 0, 3330, 5313],
    [13026, 23963, 17563, 14771, 42160, 11069, 12925, 5730, 8778, 13375, 11235, 14366, 13621, 11188, 10424, 11907, 5609, 21861, 13624, 11781, 9718, 8304, 17737, 12200, 20816, 7330, 7532, 6117, 4735, 4488, 2599, 3355, 7773, 22186, 27895, 9742, 726, 3330, 0, 2042],
    [15056, 25994, 19589, 16743, 44198, 13078, 14967, 7552, 10422, 14935, 11891, 14002, 13225, 10671, 9475, 10633, 5084, 20315, 11866, 9802, 7682, 6471, 15720, 10674, 18908, 6204, 6000, 5066, 3039, 3721, 3496, 4772, 8614, 23805, 29519, 11614, 2749, 5313, 2042, 0],
 ] # yapf: disable


def main():
    """Entry point of the program."""
    num_nodes = len(distance_matrix)
    print('Num nodes =', num_nodes)

    # Number of groups to split the nodes, must divide num_nodes.
    num_groups = 4
    group_size = num_nodes // num_groups

    # Model.
    model = cp_model.CpModel()

    # Variables.
    neighbors = {}
    obj_vars = []
    obj_coeffs = []
    for n1 in range(num_nodes - 1):
        for n2 in range(n1 + 1, num_nodes):
            same = model.NewBoolVar('neighbors_%i_%i' % (n1, n2))
            neighbors[n1, n2] = same
            obj_vars.append(same)
            obj_coeffs.append(distance_matrix[n1][n2] + distance_matrix[n2][n1])

    # Number of neighborss:
    for n in range(num_nodes):
        model.Add(sum(neighbors[m, n] for m in range(n)) + 
                  sum(neighbors[n, m] for m in range(n + 1, num_nodes)) ==
                  group_size - 1)
    
    # Enforce transivity on all triplets.
    for n1 in range(num_nodes - 2):
        for n2 in range(n1 + 1, num_nodes - 1):
            for n3 in range(n2 + 1, num_nodes):
                model.Add(
                    neighbors[n1, n3] + neighbors[n2, n3] + neighbors[n1, n2] != 2)

    # Redundant constraints on total sum of neighborss.
    model.Add(sum(obj_vars) == num_groups * group_size * (group_size - 1) // 2)

    # Minimize weighted sum of arcs.
    model.Minimize(
        sum(obj_vars[i] * obj_coeffs[i] for i in range(len(obj_vars))))

    # Solve and print out the solution.
    solver = cp_model.CpSolver()
    solver.parameters.log_search_progress = True
    solver.parameters.num_search_workers = 8

    status = solver.Solve(model)
    print(solver.ResponseStats())

    visited = set()
    for g in range(num_groups):
        for n in range(num_nodes):
            if not n in visited:
                visited.add(n)
                output = str(n)
                for o in range(n + 1, num_nodes):
                    if solver.BooleanValue(neighbors[n, o]):
                        visited.add(o)
                        output += ' ' + str(o)
                print('Group', g, ':', output)
                break


if __name__ == '__main__':
    main()