JP2000048003A - Hierarchical processing method for traveling salesman problem and its program recording medium - Google Patents

Abstract

PROBLEM TO BE SOLVED: To speed up processing and to find out a good resolution by reducing the number of areas and expanding an area capable of setting up start/end cities in a method for hierarchically solving a large scale traveling salesman problem(TSP) for searching a shortest route out of plural routes for traveling many cities scattered in a problem area on a two-dimensional plane. SOLUTION: An area division part 1 divides TSP areas and an area correction part 2 detects the deviation of cities in a certain area and corrects the area so that a part of the area having the deviation of cities is included in its adjacent area. A TSP processing part 4 regards each divided area as a city and executes TSP processing in each hierarchy. A start/end city determination part 3 determines a start/end city for executing TSP processing in each area from a previously determined settable area based on the route of a higher hierarchy at and after a 2nd hierarchy.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for solving a large-scale traveling salesman problem (hereinafter, referred to as TSP) by a computer, and more particularly to a processing method for hierarchically dividing a problem area into a solution.

[0002] The TSP is a problem of searching for a shortest route (a route with a minimum cost) among routes that go around cities scattered in a problem area on a two-dimensional plane. The number of routes to go round is (N-1) for N cities! Only exists. With the increase in the number of circulating cities, the computational complexity of city combinations increases explosively, so it is necessary to prevent the explosion of this combination, speed up the processing, and find the optimal solution. .

[0003]

2. Description of the Related Art In a conventional TSP hierarchical solution,
The processing is performed by dividing the problem area of the TSP into areas of a fixed number and a fixed size. However, in the conventional method, processing is performed assuming that an area where the arrangement of cities is biased is one area, so that the number of areas where cities exist is large, and it is necessary to perform TSP processing even in a small area. .

[0004] A hierarchical solution method of the TSP according to the conventional method will be described with reference to FIGS. FIG. 10 shows a method of dividing a city space (problem area) according to the related art. FIG.
As shown in (A), the problem area of the TSP is, for example, 4 × 4
It is divided and one area is made one city. In this example, FIG.
As shown in (B), since there are 12 areas where cities are present in the 16 areas, the TSP problem of 12 cities is processed assuming that 12 cities exist on the TSP problem area. FIG. 10C shows one solution (route) of the TSP processing in 12 cities.

[0005] Furthermore, the TSP problem is solved by similarly dividing the area by 4x4 for each area constituting the determined route.
By layering and processing in this way, the number of routes that need to be searched can be reduced.

[0006] FIG. 11 shows an example of a settable area of a start city / end city in the conventional method. In the case where the TSP process is performed in a hierarchical manner in the divided areas, it is necessary to determine a city to start a search route from and a city to end. In the conventional method, the start city and the end city are arranged in any of the four small areas adjacent to the other two areas located at the four corners of the area, as indicated by hatching in FIG.

FIG. 12 shows an example of a method of determining a start city / end city in the conventional method. In the prior art, the start city / end city is determined based on the position of the settable area of the start city / end city shown in FIG. 11 and the position of the region processed immediately before and the position of the region processed next. Determined by the positional relationship.
If there is a city in the closest installable area among the four installable areas, the installable area is set as the start city / end city.

FIG. 12A shows a case where the region processed immediately before and the region to be processed next are in the upward direction.
FIG. 12C shows a case where the immediately preceding region and the next region to be processed are in the right direction, and FIG. 12C shows a case where the immediately preceding region and the next region to be processed are in the downward direction.
(D) shows the settable area of the start city / end city when the area processed immediately before and the area processed next are in the left direction. In these cases, one possible installation area is the starting city and the other is the ending city.

FIG. 12E shows the positions of the start city and the end city when the immediately preceding area and the next area to be processed are obliquely upward. For example, if the area processed immediately before is in the upper left direction, the position of the start city is a small area in the upper left corner.

If the immediately preceding area and the next area to be processed are in the same diagonal direction, the end city and the start city can be set at the same position. In this case, One of the remaining three installable areas is set as the installable area of the end city.

If no city exists in the settable area of the start city and the end city as described above, a dummy city is arranged in the settable area for the sake of processing. FIG.
FIG. 4 is a diagram for explaining a start city / end city arrangement and hierarchical TSP processing according to a conventional method.

Each area on the route shown in FIG. 10C is further divided into 4 × 4 and TSP processing is performed. Here, attention is paid to the area X and the area Y corresponding to the city X and the city Y which are continuous on the traveling route. In the area X and the area Y, the TSP processing is further performed by further dividing into 4 × 4.

In the area X, since the area processed immediately before is the area in the upward direction, the installation position of the starting city is either the upper left or the upper right installation area. In addition, the installation position of the end city corresponds to only the installable area in the upper right because the area to be processed next is the area in the upper right direction. Accordingly, in the area X, the start city is located in the upper left installable area and the end city is located in the upper right installable area, but no city exists in either the upper left or upper right installable area. Therefore, in the upper left and upper right installation areas,
A dummy starting city and a dummy ending city are established respectively.

In the area Y, since the area processed immediately before is located in the lower left direction, the position where the starting city can be installed is only the installation area in the lower left. Further, since the area to be processed next to the area Y is a downward area, the position where the end city can be installed is the lower right installable area. However, since there is no city in the lower right corner, a dummy city is set in the lower right corner, and that is set as the end city.

[0015]

When the problem area of the TSP is divided, there are some areas in which the arrangement of cities is biased depending on the area. Compared to those that are unevenly distributed, those where cities are unevenly distributed generally have fewer cities. If such a region is recognized as one region (one city) and processing is advanced, a biased region does not necessarily have a path directly connected to a nearby region, and as a result, in the conventional technology, Solution (=
Route) is bad.

Further, if dummy cities are arranged as the start city / end city, the number of cities becomes larger than the actual number of cities, so that the number of combinations of cities also increases, and it is difficult to obtain a solution at high speed. .

If the area where the start city and the end city can be set is limited to only the small areas at the four corners of the area, the start city and the end city may be arranged in the same small area. Because the city is determined after the start city, there is a problem that the end city may not be located in a good position.

An object of the present invention is to solve the above problems and to provide means for solving a large-scale traveling salesman problem quickly and efficiently using a computer. In particular, when cities are unevenly distributed in an area in consideration of the tendency of the cities to be scattered, the conventional two areas are divided and processed as one area, thereby suppressing an increase in the number of areas and increasing the processing speed. In addition to reducing the number of cases where dummy cities need to be set up, reducing the amount of processing and improving the solution by expanding the area that can be set up in determining the start and end cities, Aim.

[0019]

The present invention has each means as shown in FIG. 1 in order to achieve the above object. FIG.
Shows a configuration example of a device for realizing the present invention.

TSP hierarchical analysis apparatus 1 for realizing the present invention
Numeral 0 includes an area dividing section 1, an area correcting section 2, a start / end city determining section 3, and a TSP processing section 4. These are composed of a computer having a CPU and a memory, and a program executed by the CPU.

The area dividing unit 1 divides the problem area of the TSP by M ×
This is a means for dividing into N. The area correcting unit 2 checks the bias of the city in each area divided by the area dividing unit 1 according to whether or not the arrangement of the city in the area corresponds to a predetermined pattern. ,
This is a means for incorporating it into an adjacent area to form one area.

The start / end city determination unit 3 is a means for determining the position of the start city or the end city, and sets a small area located at the four corners of the area and a small area adjacent to the small area as an installable area. , The city existing in it shall be the start city or end city.

The TSP processing section 4 is a means for treating the divided area as one city and performing TSP processing on the area (city). A program for realizing each of the above processing means by a computer can be stored in an appropriate recording medium such as a computer-readable portable medium memory, a semiconductor memory, and a hard disk.

The present invention operates as follows. The area dividing unit 1 divides a TSP problem area into M × N. The area correction unit 2 checks whether or not there is a city for each area divided by the area division unit 1, and if a city exists in the area, sets a flag for that area. Based on whether or not the flag pattern matches a predetermined pattern, it is determined whether or not the cities in the area are biased. If the cities in the area are biased, one of the areas in which the city exists is determined. Part in the adjacent area.

As described above, the number of areas is reduced by modifying the area to include the areas where the cities are unevenly distributed as the adjacent areas and expanding the areas.
The processing speed in the TSP processing unit 4 can be increased, and the solution can be improved because the traveling area of the neighboring city is expanded.

The start / end city determination unit 3 not only sets the small areas located at the four corners of the area as candidates for selection of the start / end city, but also selects the small areas adjacent thereto as candidates for selection of the start / end city. I do. Therefore, the probability that the dummy city is arranged becomes smaller, and it becomes possible to reduce the processing amount and obtain a good solution.

[0027]

DESCRIPTION OF THE PREFERRED EMBODIMENTS In the following, the T shown in FIG.
For the problem area of SP, TSP is hierarchically divided into 4 × 4
An example of performing the processing will be described.

FIG. 2 is a diagram showing an outline of a processing flow of the present invention. The TSP problem area is set to M
XN divisions (step S1). In this example, the area is a square with 4 × 4 division. The result of the division is shown in FIG.
Is the same as that shown in FIG.

Next, the area dividing unit 1
It is checked whether or not the cities in the area are unevenly distributed in each area divided by (step S2). When there is a bias in the city, the process proceeds to step S4, and when there is no bias, the process proceeds to step S5 (step S3).

If there is a city bias, a part of the area where the city is located is incorporated into an adjacent area according to a predetermined rule to make a new area (step S4). Specifically, the following processing is performed. If a city exists in the area, a flag is set to indicate the presence of the city in that area,
Get the flag pattern. The pattern of each area is compared with a pattern representing the bias of the city prepared in advance, and it is determined whether or not the bias of the city exists.

As shown in FIG. 3, the area is divided into M × N,
Each area is defined as (1,1), (1,2), ..., (1,
N),..., (M, 1),..., (M, N), if the city exists in a biased manner as described below, the area where the city exists is incorporated into the adjacent area. Correct the area division as follows.

1) If the city exists only in (1, i) (i = 1 to N), it is included in the left area (see FIG. 4A). 2) If the city exists only at (i, 1) (i = 1 to M), it is included in the upper area (see FIG. 4B).

3) If the city exists only in (M, i) (i = 1 to N), it is included in the right area (see FIG. 4C). 4) If the city exists only in (i, N) (i = 1 to M), it is included in the left area (see FIG. 4D).

5) The cities are (1, 1), (2, 1),
If only (1, 2) exists, it is included in the upper or left area (see FIG. 4E). 6) If the city exists only in (M, 1), (M-1, 1), and (M, 2), include it in the upper or right region (see FIG. 4 (F)).

7) The cities are (1, N), (1, N-1),
If it exists only in (2, N), it is included in the left or lower area (see FIG. 4 (G)). 8) The city is (M, N), (M-1, N), (M, N-
If it exists only in 1), it is included in the lower or right region (see FIG. 4H).

9) If the city is located in an area other than these, the area is not included in the adjacent area. FIG. 4 shows an example of a pattern for determining the bias of a city.

In FIG. 4, a small area indicated by oblique lines is an area where a city exists. It is assumed that the bias of the city in the area is eight patterns as described above. If the pattern of the flag indicating the presence of a city applies to these eight patterns, it is assumed that the cities are unevenly distributed in the area.

FIG. 5 is a diagram showing an example of an area after the area correction processing. As shown in FIG. 5, the number of areas where cities are present in the first hierarchy is 8, and in the case of the conventional method, as shown in FIG.
The P processing is completed for the eight cities. In this way, the number of regions is reduced, the number of combinations of cities is reduced, the processing speed is increased, and a good solution is obtained because the traveling region of an adjacent city is expanded.

Next, in step S5 of FIG. 2, it is checked whether the process is the first process, that is, the process of the first layer. In the case of the first processing, since the start city and the end city are determined in advance, the next step S6 is skipped. In the case of the processing of the second and subsequent hierarchies, the start city and the end city are determined by the processing described later in step S6.

Thereafter, the area is regarded as a city, and TSP processing is performed to obtain a route from the start city to the end city (step S7). The above processing from step S1 to S7 is repeated until the number of cities in the area becomes one and the area cannot be divided, or a predetermined number of layers are repeated (step S8).

Next, a method of determining a start city and an end city in this embodiment will be described. FIG. 6 shows an example of a settable area of a start city or an end city. In the prior art, only the small area at the four corners of the area as shown in FIG. In the present embodiment, the installable area is expanded to a small area at the four corners and a small area adjacent to the small area. A flag indicating that a start city and an end city can be set is set in a small area that can be set as a start city / end city. In the area composed of 5 × 5 (same for 4 × 4) small areas shown in FIG. 6, the hatched small areas are small areas with flags indicating that the start city and the end city can be set. Area. Each area also holds information on the direction of the previously processed area and the direction of the next processed area.

The start city is set by selecting one of the four installable areas based on the direction of the area processed immediately before, and the end city is similarly set in the direction of the area to be processed next. Based on this, one of the four possible installation areas is selected and installed.

As a method of selecting a start city, the processing area is divided into 2 × 2, an area closest to the position of the immediately preceding end city is selected, and a settable area within the area is selected. As a method of selecting the end city, the area to be processed next is divided into 2 × 2, and the installable area in the area closest to the boundary is selected.

If there is a small area where a city exists in the selected installable area, that small area is the start city or the end city. In other words, when there is a small area with “flag indicating installation possible area & flag indicating existence of city == 1”, the small area is a start city or an end city.

If the small area of “Flag indicating installation area & flag indicating presence of city == 1” does not exist in the selected installation area, a dummy start city or end city is set. .

When attention is paid to one of the settable areas, a large number of settable areas of the start city / end city (for example,
If there are three or more small areas), the closest one is selected. When the start city and the end city are in the same installation area (for example, when the area processed immediately before and the area to be processed next are both at the lower left), the three small installation areas are set. Select a different sub-region from the region as the start city or end city.

There are eight cases as follows in which the settable area of the start city and the settable area of the end city match. 1) The case where the previously processed area is an area located in the upper direction, and the area to be processed next is an area located in the upper left or upper right direction.

2) The case where the previously processed area is the area in the right direction and the area to be processed next is the area in the upper right or lower right direction. 3) The case where the previously processed area is the area in the downward direction and the area to be processed next is the area in the lower right or lower left direction.

4) The case where the previously processed area is the area in the left direction and the area to be processed next is the area in the upper left or lower left direction. 5) The previously processed area is an area in the upper left direction,
The next area to be processed is an area located in the upper or left direction.

6) The case where the previously processed area is the area located in the upper right direction and the area to be processed next is the area located in the right or downward direction. 7) The previously processed area is an area in the lower right direction,
The next area to be processed is an area located to the right or downward.

8) The case where the previously processed area is the area located in the lower left direction, and the area to be processed next is the area located in the left or lower direction. As described above, when the installable areas where the start city and the end city are set match, the start city and the end city are determined as follows.

1) When there are two or more cities in the settable area of the start city and the end city, the start city and the end city are selected from the settable area. 2) When there is only one city in the settable area of the start city and the end city, either the start city or the end city is selected from the settable area, and the remaining start or end cities are dummy cities.

3) If there is no city in the settable area of the start city and the end city, a dummy city is set as the start city and the end city. If the settable areas for setting the start city / end city do not match, the start city or the end city is selected from the settable areas.

FIG. 7 is a diagram for explaining an example of the arrangement of the start city and the end city. In FIG. 7, the region correction process is omitted in order to make it easier to understand the process of determining the start city / end city.

As shown in FIG. 7, in the area a, the previously processed area is the area located in the upper direction, and the area to be processed next is the area located in the upper right direction. The area is the upper left, and the area where the end city can be installed is the upper right. As for this area a, since a city exists in the settable area of the start city / end city and one small area in which the city exists in this settable area, the small area is set to the start city or the end city. It becomes a city, and TSP processing can be performed without setting the start city / end city of the dummy.

As for the area b, the area processed immediately before is the area located in the lower left direction, and the area to be processed next is the area located in the downward direction. Therefore, the settable area of the start city is selected from the lower left, and the settable area of the end city is selected from the lower left or lower right. In the selection of the end city installation possible area, the city at the lower left is selected because the city closest to the boundary is at the lower left. In this case, both the starting city and the ending city are selected from the three small areas of the small area at the lower left corner and the adjacent small area. here,
If the route intersects, the solution will be bad in the hierarchy. To prevent the route from intersecting, set the starting city in the adjacent small area above the lower left corner, and set the ending city in the small area in the lower left corner. Install.

FIG. 8 is a view for explaining the arrangement of the start city and the end city after the area correction processing and the hierarchical TSP processing. If the area where the cities are unevenly distributed is examined and the area is corrected as shown in FIG. 5, in the first hierarchical TSP processing, a route is searched by regarding eight areas as cities. As a result, a route between regions as shown in FIG. 8A is determined. Next, in each area on this route, further area division is performed, and the TSP processing is executed hierarchically.

A better solution is obtained when the divided area is as square as possible. However, when the area correction processing has already been performed, the area corrected due to the bias of the city, such as the area c, is no longer a square. Here, if a small area is to be created by 4 × 4 division similarly to the other uncorrected area d, the side in the direction in which a part of the adjacent area is incorporated in the area correction processing becomes longer, and the area becomes longer. The shape gradually moves away from the square with the stratification. On the other hand, if the shape of the small area is to be kept as a square, the number of areas, that is, the number of cities increases, and the processing speed decreases.

Then, for the already corrected area,
When a small area is generated by the division processing, the rectangular area is divided on the basis of the long side of the rectangle, and the divided area is divided into a rectangle as close to a square as possible. In the case of this example, if the division is made so that the size of the divided small area is the same as the small area of the area d, the division becomes 5 × 4, but the number of small areas is 2
As shown in FIG. 8 (B), the long side of the rectangle is divided into four parts, and the short side of the rectangle is divided into three parts, as shown in FIG. It does not exceed 16.

FIG. 9 is a diagram showing a configuration example of the area table. The area table holds, for each area divided in the same hierarchy, information such as the position of the area, the actual number of cities included in the area, a pointer to the next city, a pointer to the next area, and the like.

For example, in the area of the first hierarchy, links are made using pointer information to the next area in accordance with the order of the paths obtained by the TSP processing. Modify the link by changing the pointer whenever the route changes.

Further, for each area, T
The order from the start city to the end city of the route of the area (city) subjected to the SP processing is stored by a pointer or the like. Further, when the TSP processing is performed in the lower hierarchy, similarly, the route information of the city in the lower hierarchy is stored for each area by a pointer. As a result, even if the hierarchies of processing are advanced, it is possible to obtain routes for all the cities, that is, solutions, by sequentially following the pointers of the respective areas. The configuration of the area table in FIG. 9 is an example, and the area and city information may be managed in another format.

[0063]

As described above, according to the present invention,
In the Large-Scale Traveling Salesman Problem (TSP), by considering the bias of cities due to area division and improving the pattern of determining the start city and end city in the hierarchical processing of TSP,
The traveling salesman problem can be solved at high speed, and a good solution can be obtained.

[Brief description of the drawings]

FIG. 1 is a diagram showing a configuration example of an apparatus for realizing the present invention.

FIG. 2 is a diagram showing an outline of a processing flow of the present invention.

FIG. 3 is a diagram for explaining a process of correcting a divided area due to a bias of a city;

FIG. 4 is a diagram illustrating an example of a pattern for determining a bias of a city;

FIG. 5 is a diagram illustrating an example of an area after an area correction process.

FIG. 6 is a diagram showing an example of a settable area of a start city / end city.

FIG. 7 is a diagram illustrating an example of an arrangement of a start city and an end city.

FIG. 8 is a diagram illustrating the arrangement of a start city and an end city after a region correction process and a hierarchical TSP process.

FIG. 9 is a diagram illustrating a configuration example of an area table.

FIG. 10 is a diagram illustrating a method of dividing a city space (problem area) in a conventional method.

FIG. 11 is a diagram showing an installation possible area of a start city / end city in the conventional method.

FIG. 12 is a diagram showing an example of a method of determining a start city / end city in a conventional method.

FIG. 13 is a diagram illustrating the arrangement of a start city / end city and hierarchical TSP processing according to a conventional method.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Area division part 2 Area correction part 3 Start / end city determination part 4 TSP processing part 10 TSP hierarchical analysis device

Claims (4)

[Claims] 1. A processing method for hierarchically solving a traveling salesman problem in which a computer searches for the shortest route among routes traversing a large number of cities scattered in a problem area on a two-dimensional plane, the processing method comprising: Is divided into a plurality of regions.
Investigate the process and the bias of the city in each of the areas divided into the multiple areas. If the cities in the area are unevenly distributed, correct the area division so that the adjacent areas on the unevenly distributed side include those cities. And a third step of performing a process of searching for the shortest route among the routes that go around the city by regarding the divided area as a city,
A traveling salesman problem hierarchical processing method, wherein the third step is repeatedly performed hierarchically on the divided areas. 2. The traveling salesman problem hierarchical processing method according to claim 1, wherein when the positions of the start city and the end city are determined for the search in the third step, the area to be searched is determined. If the starting and ending cities are selected from the sub-regions that are located at the four corners and the sub-regions that are adjacent to the sub-regions in the sub-regions obtained by further dividing the sub-regions, and no cities exist in those sub-regions (2) A traveling salesman problem hierarchical processing method characterized in that dummy cities are arranged in these small areas and set as start cities or end cities. 3. A program for causing a computer to execute a process of layering and solving a traveling salesman problem for searching for the shortest route among routes circulating in a number of scattered cities in a problem area on a two-dimensional plane is recorded. A medium for performing a first process of dividing the problem area into a plurality of areas and examining a bias of cities in each of the plurality of areas; Modify the division of the area so that the neighboring areas of the cities include those cities.
And a third process of searching for the shortest route among the routes that go around the city by regarding the divided region as a city, and the first to third processes are divided. A traveling salesman problem hierarchical processing program recording medium, characterized by recording a program for causing a computer to perform a process of repeatedly executing a region in a hierarchical manner. 4. The traveling salesman problem hierarchical processing program recording medium according to claim 3, wherein when the positions of a start city and an end city are determined for the search in the third process, the search is performed. Of the sub-areas into which the area to be subdivided is further divided, the start city and the end city are selected from the sub-areas located at the four corners and the cities present in the sub-areas bordering the sub-areas. If there is no such program, a program for recording a program for causing a computer to execute a process of arranging dummy cities in those small areas and setting the city as a start city or an end city is recorded. recoding media.