pgr_TSP

  • pgr_TSP - Aproximation using metric algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability:

  • Version 3.2.1

    • Metric Algorithm from Boost library

    • Simulated Annealing Algorithm no longer supported

      • The Simulated Annealing Algorithm related parameters are ignored: max_processing_time, tries_per_temperature, max_changes_per_temperature, max_consecutive_non_changes, initial_temperature, final_temperature, cooling_factor, randomize

  • Version 2.3.0

    • Signature change

      • Old signature no longer supported

  • Version 2.0.0

    • Official function

Description

Problem Definition

The travelling salesperson problem (TSP) asks the following question:

Given a list of cities and the distances between each pair of cities, which is the shortest possible route that visits each city exactly once and returns to the origin city?

Characteristics

  • This problem is an NP-hard optimization problem.

  • Metric Algorithm is used

  • Implementation generates solutions that are twice as long as the optimal tour in the worst case when:

    • Graph is undirected

    • Graph is fully connected

    • Graph where traveling costs on edges obey the triangle inequality.

  • On an undirected graph:

    • The traveling costs are symmetric:

    • Traveling costs from u to v are just as much as traveling from v to u

  • Can be Used with Cost Matrix - Category functions preferably with directed => false.

    • With directed => false

      • Will generate a graph that:

        • is undirected

        • is fully connected (As long as the graph has one component)

        • all traveling costs on edges obey the triangle inequality.

      • When start_vid = 0 OR end_vid = 0

        • The solutions generated is garanteed to be twice as long as the optimal tour in the worst case

      • When start_vid != 0 AND end_vid != 0 AND start_vid != end_vid

        • It is not garanteed that the solution will be, in the worse case, twice as long as the optimal tour, due to the fact that end_vid is forced to be in a fixed position.

    • With directed => true

      • It is not garanteed that the solution will be, in the worse case, twice as long as the optimal tour

      • Will generate a graph that:

        • is directed

        • is fully connected (As long as the graph has one component)

        • some (or all) traveling costs on edges might not obey the triangle inequality.

      • As an undirected graph is required, the directed graph is transformed as follows:

        • edges (u, v) and (v, u) is considered to be the same edge (denoted (u, v)

        • if agg_cost differs between one or more instances of edge (u, v)

        • The minimum value of the agg_cost all instances of edge (u, v) is going to be considered as the agg_cost of edge (u, v)

        • Some (or all) traveling costs on edges will still might not obey the triangle inequality.

  • When the data is incomplete, but it is a connected graph:

    • the missing values will be calculated with dijkstra algorithm.

Signatures

Summary

pgr_TSP(Matrix SQL, [start_id], [end_id])
RETURNS SETOF (seq, node, cost, agg_cost)
Example:

Using pgr_dijkstraCostMatrix to generate the matrix information

  • Line 4 Vertices \(\{2, 4, 13, 14\}\) are not included because they are not connected.

 1SELECT * FROM pgr_TSP(
 2  $$SELECT * FROM pgr_dijkstraCostMatrix(
 3    'SELECT id, source, target, cost, reverse_cost FROM edges',
 4    (SELECT array_agg(id) FROM vertices WHERE id NOT IN (2, 4, 13, 14)),
 5    directed => false) $$);
 6 seq | node | cost | agg_cost
 7-----+------+------+----------
 8   1 |    1 |    0 |        0
 9   2 |    3 |    1 |        1
10   3 |    7 |    1 |        2
11   4 |    6 |    1 |        3
12   5 |    5 |    1 |        4
13   6 |   10 |    2 |        6
14   7 |   11 |    1 |        7
15   8 |   12 |    1 |        8
16   9 |   16 |    2 |       10
17  10 |   15 |    1 |       11
18  11 |   17 |    2 |       13
19  12 |    9 |    3 |       16
20  13 |    8 |    1 |       17
21  14 |    1 |    3 |       20
22(14 rows)
23

Parameters

Parameter

Type

Description

Matrix SQL

TEXT

Matrix SQL as described below

TSP optional parameters

Column

Type

Default

Description

start_id

ANY-INTEGER

0

The first visiting vertex

  • When 0 any vertex can become the first visiting vertex.

end_id

ANY-INTEGER

0

Last visiting vertex before returning to start_vid.

  • When 0 any vertex can become the last visiting vertex before returning to start_id.

  • When NOT 0 and start_id = 0 then it is the first and last vertex

Inner Queries

Matrix SQL

Matrix SQL: an SQL query, which should return a set of rows with the following columns:

Column

Type

Description

start_vid

ANY-INTEGER

Identifier of the starting vertex.

end_vid

ANY-INTEGER

Identifier of the ending vertex.

agg_cost

ANY-NUMERICAL

Cost for going from start_vid to end_vid

Result Columns

Returns SET OF (seq, node, cost, agg_cost)

Column

Type

Description

seq

INTEGER

Row sequence.

node

BIGINT

Identifier of the node/coordinate/point.

cost

FLOAT

Cost to traverse from the current node to the next node in the path sequence.

  • 0 for the last row in the tour sequence.

agg_cost

FLOAT

Aggregate cost from the node at seq = 1 to the current node.

  • 0 for the first row in the tour sequence.

Additional Examples

Start from vertex \(1\)

  • Line 6 start_vid => 1

 1SELECT * FROM pgr_TSP(
 2  $$SELECT * FROM pgr_dijkstraCostMatrix(
 3    'SELECT id, source, target, cost, reverse_cost FROM edges',
 4    (SELECT array_agg(id) FROM vertices WHERE id NOT IN (2, 4, 13, 14)),
 5    directed => false) $$,
 6  start_id => 1);
 7 seq | node | cost | agg_cost
 8-----+------+------+----------
 9   1 |    1 |    0 |        0
10   2 |    3 |    1 |        1
11   3 |    7 |    1 |        2
12   4 |    6 |    1 |        3
13   5 |    5 |    1 |        4
14   6 |   10 |    2 |        6
15   7 |   11 |    1 |        7
16   8 |   12 |    1 |        8
17   9 |   16 |    2 |       10
18  10 |   15 |    1 |       11
19  11 |   17 |    2 |       13
20  12 |    9 |    3 |       16
21  13 |    8 |    1 |       17
22  14 |    1 |    3 |       20
23(14 rows)
24

Using points of interest to generate an asymetric matrix.

To generate an asymmetric matrix:

  • Line 4 The side information of pointsOfInterset is ignored by not including it in the query

  • Line 6 Generating an asymetric matrix with directed => true

    • \(min(agg\_cost(u, v), agg\_cost(v, u))\) is going to be considered as the agg_cost

    • The solution that can be larger than twice as long as the optimal tour because:

      • Triangle inequality might not be satisfied.

      • start_id != 0 AND end_id != 0

 1SELECT * FROM pgr_TSP(
 2  $$SELECT * FROM pgr_withPointsCostMatrix(
 3    'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
 4    'SELECT pid, edge_id, fraction from pointsOfInterest',
 5    array[-1, 10, 7, 11, -6],
 6    directed => true) $$,
 7  start_id => 7,
 8  end_id => 11);
 9 seq | node | cost | agg_cost
10-----+------+------+----------
11   1 |    7 |    0 |        0
12   2 |   -6 |  0.3 |      0.3
13   3 |   -1 |  1.3 |      1.6
14   4 |   10 |  1.6 |      3.2
15   5 |   11 |    1 |      4.2
16   6 |    7 |    1 |      5.2
17(6 rows)
18

Connected incomplete data

Using selected edges \(\{2, 4, 5, 8, 9, 15\}\) the matrix is not complete.

 1SELECT * FROM pgr_dijkstraCostMatrix(
 2  $q1$SELECT id, source, target, cost, reverse_cost FROM edges WHERE id IN (2, 4, 5, 8, 9, 15)$q1$,
 3  (SELECT ARRAY[6, 7, 10, 11, 16, 17]),
 4  directed => true);
 5 start_vid | end_vid | agg_cost
 6-----------+---------+----------
 7         6 |       7 |        1
 8         6 |      11 |        2
 9         6 |      16 |        3
10         6 |      17 |        4
11         7 |       6 |        1
12         7 |      11 |        1
13         7 |      16 |        2
14         7 |      17 |        3
15        10 |       6 |        1
16        10 |       7 |        2
17        10 |      11 |        1
18        10 |      16 |        2
19        10 |      17 |        3
20        11 |       6 |        2
21        11 |       7 |        1
22        11 |      16 |        1
23        11 |      17 |        2
24        16 |       6 |        3
25        16 |       7 |        2
26        16 |      11 |        1
27        16 |      17 |        1
28        17 |       6 |        4
29        17 |       7 |        3
30        17 |      11 |        2
31        17 |      16 |        1
32(25 rows)
33

Cost value for \(17 \rightarrow 10\) do not exist on the matrix, but the value used is taken from \(10 \rightarrow 17\).

 1SELECT * FROM pgr_TSP(
 2  $$SELECT * FROM pgr_dijkstraCostMatrix(
 3  $q1$SELECT id, source, target, cost, reverse_cost FROM edges WHERE id IN (2, 4, 5, 8, 9, 15)$q1$,
 4  (SELECT ARRAY[6, 7, 10, 11, 16, 17]),
 5  directed => true)$$);
 6 seq | node | cost | agg_cost
 7-----+------+------+----------
 8   1 |    6 |    0 |        0
 9   2 |    7 |    1 |        1
10   3 |   11 |    1 |        2
11   4 |   16 |    1 |        3
12   5 |   17 |    1 |        4
13   6 |   10 |    3 |        7
14   7 |    6 |    1 |        8
15(7 rows)
16

See Also

Indices and tables