Linear Programming in Python

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from pulp import *
from itertools import *

# graphs as an adjacency list - how far is it from vertex i to vertex j
# taken from https://people.sc.fsu.edu/~jburkardt/datasets/tsp/tsp.html
graphs = [
    [
        "FIVE",
        [[0], [3, 0], [4, 4, 0], [2, 6, 5, 0], [7, 3, 8, 6, 0]],
    ],
    [
        "P01",
        [
            [0],
            [29, 0],
            [82, 55, 0],
            [46, 46, 68, 0],
            [68, 42, 46, 82, 0],
            [52, 43, 55, 15, 74, 0],
            [72, 43, 23, 72, 23, 61, 0],
            [42, 23, 43, 31, 52, 23, 42, 0],
            [51, 23, 41, 62, 21, 55, 23, 33, 0],
            [55, 31, 29, 42, 46, 31, 31, 15, 29, 0],
            [29, 41, 79, 21, 82, 33, 77, 37, 62, 51, 0],
            [74, 51, 21, 51, 58, 37, 37, 33, 46, 21, 65, 0],
            [23, 11, 64, 51, 46, 51, 51, 33, 29, 41, 42, 61, 0],
            [72, 52, 31, 43, 65, 29, 46, 31, 51, 23, 59, 11, 62, 0],
            [46, 21, 51, 64, 23, 59, 33, 37, 11, 37, 61, 55, 23, 59, 0],
        ],
    ],
    [
        "GR17",
        [
            [0],
            [633, 0],
            [257, 390, 0],
            [91, 661, 228, 0],
            [412, 227, 169, 383, 0],
            [150, 488, 112, 120, 267, 0],
            [80, 572, 196, 77, 351, 63, 0],
            [134, 530, 154, 105, 309, 34, 29, 0],
            [259, 555, 372, 175, 338, 264, 232, 249, 0],
            [505, 289, 262, 476, 196, 360, 444, 402, 495, 0],
            [353, 282, 110, 324, 61, 208, 292, 250, 352, 154, 0],
            [324, 638, 437, 240, 421, 329, 297, 314, 95, 578, 435, 0],
            [70, 567, 191, 27, 346, 83, 47, 68, 189, 439, 287, 254, 0],
            [211, 466, 74, 182, 243, 105, 150, 108, 326, 336, 184, 391, 145, 0],
            [268, 420, 53, 239, 199, 123, 207, 165, 383, 240, 140, 448, 202, 57, 0],
            [
                246,
                745,
                472,
                237,
                528,
                364,
                332,
                349,
                202,
                685,
                542,
                157,
                289,
                426,
                483,
                0,
            ],
            [
                121,
                518,
                142,
                84,
                297,
                35,
                29,
                36,
                236,
                390,
                238,
                301,
                55,
                96,
                153,
                336,
                0,
            ],
        ],
    ],
]

for name, distances in graphs:
    n = len(distances)

    model = LpProblem(sense=LpMinimize)

    # variables - binary x_{i, j} based on if the edge (i, j) is on the cycle
    # they're symmetric so we only care about i > j
    # variables x_{i, i} will always be zero, they're there for nicer code
    variables = [
        [LpVariable(name=f"x_{i}_{j}", cat=LpBinary) for j in range(i + 1)]
        for i in range(n)
    ]

    # inequalities
    ## each vertex has one incoming and one outgoing edge
    for i in range(n):
        model += lpSum([variables[max(i, j)][min(i, j)] for j in range(n)]) == 2
        model += variables[i][i] == 0

    ## each subset of vertices has an outgoing edge (no smaller loops!)
    for size in range(1, n - 1):
        for subset in combinations(range(n), r=size):
            model += (
                lpSum(
                    [
                        variables[max(i, j)][min(i, j)]
                        for i in subset
                        for j in range(n)
                        if j not in subset
                    ]
                )
                >= 1
            )

    # objective function - minimize the length of the cycle
    model += lpSum(
        [
            variables[i][j] * distances[i][j]
            for i, j in product(range(n), repeat=2)
            if i > j
        ]
    )

    status = model.solve(PULP_CBC_CMD(msg=False))

    print(name + "\n" + "-" * len(name))
    print("minimal tour: ", int(model.objective.value()))
    for i in range(n):
        for j in range(n):
            print(int(variables[max(i, j)][min(i, j)].value()), " ", end="")
        print()
    print()