The Traveling Salesman Problem New Mexico Supercomputing Challenge Final Report April 6 th, 2016 Team 104 School of Dreams Academy

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1 The Traveling Salesman Problem New Mexico Supercomputing Challenge Final Report April 6 th, 2016 Team 104 School of Dreams Academy Team Members: Victoria Troyer Sponsor: Kerra Howe Mentor: Zack Daniels 0

2 Table of contents Background 2 Introduction.3 Programming Method...3 Hypothesis...5 Experimental Procedure...5 Results...7 Discussion.14 Conclusion 15 Practical Application 16 Future Expansion..17 Acknowledgments.17 References.18 1

3 Background The Traveling Salesman Problem (TSP) is a mathematical approach to the following question: Given a series of points and the distances between these points, what is the shortest possible route that visits each point exactly once and returns to the point of origin? The simplicity of the statement of the problem is deceptive -- the TSP is one of the most intensely studied problems in computational mathematics and yet no effective solution method is known for the general case. Although the complexity of the TSP is still unknown, for over 50 years its study has led the way to improved solution methods in many areas of mathematical optimization. [1] The history of research into the TSP dates back to at least [2] Companies such as United Parcel Service and FedEx use an algorithm similar to the Traveling Salesman to find the shortest route to deliver packages. The problem these companies have with programs utilized to find a reasonable route is that their algorithms are complex and time intensive. Due to the time factor of these programs, users do not typically wait for the programs to solve for the shortest route. Delivery companies often find a reasonable route as opposed to the shortest route using Heuristics. [2] Some researchers believe that the TSP is in a class of problems known as NP-Hard meaning that these problems contain nondeterministic polynomials and may not have any solution and that approximations through heuristics are the only solutions possible other than through the Brute Force Method. [3] 2

4 Introduction. The purpose of this project is to test alternative solutions to the Traveling Salesman Problem (TSP): the Brute Force Method and by solving using different grouping methods. The complexity of these different solutions vary. The Brute Force Method is the most basic, while grouping methods are more complex. [4] The Traveling Salesman Problem is so resource intensive that grouping approaches have a good probability of improving solution times. The basic Brute Force algorithm is utilized in all of the tested TSP programs, however the biggest difference between the programs is what size of data set the algorithm is allowed to process. As the data set grows the Brute Force algorithm requires more and more computing resources and time. This project explores the possibility of using different techniques to contain data sets to smaller sizes that are easier for the Brute Force algorithm to process. The basic Brute Force Method is used as a control to test the efficiency of the Grouping and Quadrant Methods. The Grouping Method finds small closely spaced points on the map then groups these together and performs the Brute Force algorithm on these smaller groups before moving on to define and process the next small group. The Quadrant Method is similar to the Grouping Method in that it breaks the total number of points into smaller groups; however, the groups are defined by dividing the map into equally sized quadrants and then processing the points within these quadrants using the Brute Force algorithm. Programming Methods 3

5 The Brute Force Method works by focusing on one given point generated on the interface. Knowing the exact location of each point, it records the distance from the given point to all other points. It then records this data into a list called distance list. The program then looks through all the recorded distances and finds the distance with the smallest value. This value represents the shortest path traveled from one point to the other. After this process is repeated with every given point, all steps to solve for the shortest route are complete. The shortest route is then traveled by the agent and represented as a red line that visits each point once. The issue with this program is that it can only solve the shortest route for a maximum limit of eight points, because there is so much data being processed the program will crash above this number. Since the Brute Force program was restricted to a max of eight points, the alternative solution to this problem was separate the points into a series of groups then run the program a multitude of times. This allowed the program to solve for the shortest route of a greater amount of points. This is essentially how the Grouping and Quadrants Methods work, though they contrast in the different methods employed to group the points. The Quadrants Method works by setting up the interface much like a grid by dividing the map into four equally sized quadrants. The program then solves the shortest route for all points generated in quadrant one. As the agent travels the shortest route for quadrant one, it moves all other visited points to the location of the last visited point. This processes is then repeated for all other quadrants. After the points have all been moved to a single location for each quadrant, the Brute Force Method is used again to solve the final shortest route, though this was an issue when generating a random number of points in random locations. This is because if there happened to be over eight points generated in a given quadrant the program would crash, which is why the grouping method was developed. 4

6 The Grouping Method has a different approach for separating the points. This method looks at the location of the agent, and finds a number of the closest points from the agent and labels them as group one. The program then solves the shortest route using Brute Force method. Again, as the program travels the shortest route for group one it moves all other visited points to the location of the last visited point. From the location of that ending point in group one the process is then repeated. After the points have all been moved to one location for each group, the Brute Force Method is used again to solve the final shortest route. Hypothesis The three approaches used to solve the TSP were the Brute Force Method, Quadrant Method, and Grouping Method. The Brute Force Method was expected to reliably solve the problem, however, the large data set created by this method meant that it would take a very long time to produce a solution. The Quadrant Method was expected to have similar but slightly longer times than the Grouping Method. The Grouping Method was expected to solve for the most efficient route in the shortest amount of time. Experimental Procedure 1. The first step for this project was develop a simple program code for solving the TSP, this program would eventually be named the Brute Force Method. 2. Determine what portions of the code are inefficient and which aspects of the program code are required to produce shortest route solutions. While keeping the useful aspects the program code was modified by producing the following effects for each alternative solution method. 3. Develop three distinct approaches to the shortest route solution. 5

7 a. Brute Force Method: i. Measure and tabulate distances between every single point to every single other point. ii. Sums the total distance for every possible route. This will produce iii. Finds the shortest route solution produced by beginning the route with each point. iv. Assemble the final route to all points. v. Calculates total distance traveled and the time taken to produce the shortest route solution. b. Quadrant Method: i. Split the map into equal quadrants ii. Performs Brute Force calculations on all points located in each quadrant one quadrant at a time. iii. Assembles the final route to all points. iv. Calculates total distance traveled and the time taken to produce the shortest route solution. c. Grouping Method i. Grouping points that are close together. ii. Performing Brute Force calculations on each group of points before moving to the next group points. iii. Moving from one group to the next. iv. Assembling the final route to among all points. 6

8 v. Calculating the overall distance traveled and the time taken to produce the shortest route solution. Results The results of this project are shown through graphs and charts below. The Brute Force Method had average solution times of more than 25 seconds for all maps tested, all maps tested had only eight points randomly placed on each map. The average solution times for the Brute Force Method are shown in the bar graph in figure one below. The Brute Force Method s long solution times are characteristic of the inefficient processing of large data sets. The Quadrants Method and Grouping Method both took less than one tenth of a second to produce solutions. The bar graph shown in figure 2 indicates that the Quadrant method is a vast improvement over the Brute Force Method being a thousand times faster with average solution times at or below thirty thousandths of a second. The Grouping Method s average solution times were even more efficient being ten times faster than the Quadrants Method. The high level of efficiency meant that maps with sixteen points could be solved quickly by the Quadrants and Grouping Methods in contrast to the limited eight point maps used to test the Brute Force Method. 7

9 Average Time (ms) Average Time Brute Force Random Seed Figure 1. Brute Force Method took longer than 25 seconds to solve maps with only 8 randomly placed points. Quadrants vs. Groups Random Seed Quads Groups Figure 2. Solution times for Quadrant and Grouping Methods took less than one second to solve for maps with sixteen randomly placed points. 8

10 Figures 3 through 6 are line graphs showing the actual solutions times of the Grouping Method s test runs. Each graph contains 1000 points of solution time data for a single map with sixteen randomly placed points. Also plotted on the graph are the lines which represents the average solution time for all 1000 runs (the bottom most straight line) as well as the 1, 2, and 3 Sigma lines which represent the average plus 1, 2,and 3 standard deviations respectively (the three upper straight lines). These straight lines are often used by engineers to monitor and control processes. Data points which fall outside of the 3 Sigma line are considered to be anomalous. In this case the 3 Sigma line is used to indicate how far away from the average the plotted data points are, in this way we can better define the difference between outlier and anomalies among the data points. All solution times greater than 3 Sigma are considered to be anomalies while outlier data points are considered to be those between the 3 Sigma line and the Average line. Due to the number of anomalies and their high values the average is somewhat skewed toward the high end of the data range, however we use the fact that the average is skewed high to define the lower limit for outlier points. It should be noted that while the collected data does include anomalies and outliers the occurrence of the anomalies at about seconds is consistent among all maps tested. Average solution times are also fairly consistent among all the maps falling between and seconds for Grouping Method trials. 9

11 Solution Time (seconds) Solution Time (seconds) Grouping Map Run Time Average 1 Sigma 2 Sigma 3 Sigma Figure 3. Grouping Map 5, Std. Dev=0.0053, 2.1 % of all runs took more than seconds (above 3 Sigma) to solve and are considered anomalous, 97.8 % of all solutions were within one sigma. Max solve time=0.046 s, min solve time=0.001 s Grpouping Map Time Average 1 Sigma 2 Sigma 3 Sigma Run Figure 4.. Grouping Map 6, Std. Dev=0.0041, 1.5 % of all runs took more than seconds (above 3 Sigma) to solve,98.2% of all solutions were within one sigma. Max solve time=0.035 s, min solve time=0.001 s 10

12 Solution Time (seconds) Solution Time (seconds) Grouping Map Time Average 1 Sigma 2 Sigma 3 Sigma Run Figure 5. Grouping Map 7, Std. Dev=0.0045, 1 % of all runs took more than seconds (above 3 Sigma) to solve, 98% of all solutions were within one sigma. Max solve time=0.035 s, min solve time=0.001 s Grouping Map Time Average 1 Sigma 2 Sigma 3 Sigma Run Figure 6. Grouping Map 8, Std. Dev=0.0045, 1.3 % of all runs took more than seconds (above 3 Sigma) to solve, 98.3 % of all solutions were within one sigma. Max solve time=0.049 s, min solve time=

13 Solution Time (seconds) Figures 7 through 10 are the line graphs for the Quadrants Method and these graphs are formatted the same as those above for the Grouping Method. Plotted with the data are the Average line and also the 1, 2, and 3 Sigma lines (Average plus 1, 2, and 3 standard deviations respectively). The most notable differences between these graphs for the Quadrants and the previous graphs for the Grouping Method are that in only figure 7 the anomalous data (above 3 Sigma) is very similar to that of the Grouping Method data, a fact which we cannot explain at present and conversely the graphs in figures 8, 9, and 10 are very similar to one another but not at all similar to that in figure 7, another unexplained fact. One possible reason for these similarities of data graphs for Quadrant Map 5 to all of those for Grouping Method may be that the random distribution of points on the map fell into locations which caused a solution pattern which closely approximated that of the Grouping Method, however; the random distribution of points for the remaining maps produced no such similar solution pattern and instead produced solution patterns uniquely characteristic of the Quadrant Method Quadrants Map Time Average 1 Sigma 2 Sigma 3 Sigma Run Figure 7. Quadrant Map 5, Std. Dev= , 3.3 % of all runs took more than 0.13 seconds (above 3 Sigma) to solve, 96.7% of all solutions were within one sigma. Max solve time=0.24 s, min solve time=0.02 s 12

14 Solution Time (seconds) Solution Tme (seconds) Quadrants Map Time Average 1 Sigma 2 Sigma 3 Sigma Run Number Figure8. Quadrants Map 6, Std. Dev = ,.6 % of all runs took more than seconds (above 3 Sigma) to solve,98.2% of all solutions were within one sigma. Max solve time=0.075 s, min solve time=0.012 s 0.07 Quadrants Map Timer Average 1 Sigma 2 Sigma 3 Sigma Run Figure 9. Quadrants Map 7, Std. Dev=0.0047,.6 % of all runs took more than 0.04 seconds (above 3 Sigma) to solve, 98.2% of all solutions were within one sigma. Max solve time=0.065 s, min solve time=0.012 s 13

15 Solution Time (seconds) Quadrants Map Timer Average 1 Sigma 2 Sigma 3 Sigma Run Figure 10. Quadrants Map 8, Std. Dev=0.0042,.6 % of all runs took > seconds to solve,98.2% of all solutions were within one sigma. 0.3% of solutions times can be considered outliers. Max solve time=0.069 s, min solve time=0.024 s Discussion The charts above were unexpected when the data were first graphed. Initially it was thought that there was a problem when the data was collected making it unreliable. It wasn t until further investigation that it was realized the cause of the extremely high peaks in the charts were due to the process of outputting data statistics and graphical information to the computer screen during program runtime. Brute Force Method is time consuming and inefficient, by tabulating all possible routes between all points on the map a great deal of time and computing resources are required to produce the shortest route solution. The total number of routes calculated in this manner is equal to the factorial function: Number of Possible Solutions = (n-1)!/2 14

16 where n is the number of points on the map and the -1 represents the starting point and the /2 eliminates half of the possibilities as mirrored routes identical in length to another solution which traverses all the same points in the reverse order. [2] The Brute Force program was much less efficient than either Quadrants or Grouping programs. Brute Force taking at least twenty-five seconds (25 seconds) to solve maps with only eight randomly placed points which is a thousand times longer than Quadrants twenty-five thousandths of a second (0.025 seconds) to solve for maps with twice as many randomly placed points. Grouping was the most efficient method taking at most twenty-six ten-thousandths of a second ( seconds) to solve sixteen point maps identical to those tested on the Quadrant method. Conclusion By employing different methods of data parsing it is possible to greatly increase the efficiency of a traveling salesman program. From Brute Force to the Quadrants method, more than three orders of magnitude were gained, and again another order of magnitude was gained over the Quadrants method by the Grouping method. The computer is able to faster process the smaller data sets during program execution by breaking the points on the map into smaller groups and then applying the basic algorithm that examines all possible routes in the group to find the shortest path. This is the essence of a computational algorithm; a set of simple steps which together produce a solution to the problem. [5] 15

17 Practical Application The traveling salesman program can be applied practically to many different problems. It can be used to design circuit boards and program robotic machinery. [6] It is often used to analyze biomedical phenomena such as bacterial growth and the spreading of diseases as well as many other fields of study in this case, however; we mainly consider a delivery truck route as mentioned in the introduction of this report. The Brute Force method is the basic algorithm for solving the TSP but, while simple; it is inefficient in processing large data sets often making it practically useless due to its disregard for real world scenarios of naturally grouped points on a map. [4] The Grouping Method takes direct advantage of the real world fact that: points are often grouped together by shorter distances and associated with one or two common nodes. For example: a gated entrance to a subdivision might be considered a node for a delivery truck driver because it is the common access point for all houses within that particular subdivision, and the driver must use that leg of his route at least twice; once to enter the subdivision and again to exit. Other examples of grouping applications include farming tasks such as piling rocks, stacking hay-bales or gathering produce. [9] The Quadrant method may be more suitable for specific situations in which the points are naturally divided into quadrants geographically. Many large cities are organized into quadrants and travel distances between adjacent quadrants is greater than short distances between points within a quadrant. All points within each quadrant should be visited before moving to the next quadrant and transitioning between those quadrants represented by a longer leg in the overall route and so the frequency of use of those longer legs should be minimized for efficiency. Finally a combination of all these approaches may be the most efficient where the map being analyzed has some aspect of all these features. 16

18 Future Expansion My mentors and I envision an improved program capable of recognizing certain features of a map that are amenable to processing by one or a more grouping methods then adapts the processing approach to that portion of the map and analyzes and processes other portions of the map separately but in similar fashion until all points on the map have been processed and the final result is found. A program that is able to determine the best processing approach and also use multiple combinations of approaches on a given map may be most versatile, however; increasing the complexity of the program may cause a drop in efficiency. While some trade-offs may be appropriate, these should be made with care and a thorough analysis of any decrease in efficiency is also necessary. Acknowledgements Many thanks to all who made this research possible. Zack Daniels-Student Mentor Creighton Eddington-Mentor Lawrence Johnson-Mentor Kerra Howe-Sponsor 17

19 References 1. "The Problem." The Problem. Web. 22 Mar "Feature Column from the AMS." American Mathematical Society. Web. 05 Apr Weisstein, Eric W. "NP-Hard Problem." From MathWorld--A Wolfram Web Resource "13. Case Study: Solving the Traveling Salesman Problem." 13. Case Study: Solving the Traveling Salesman Problem CMPT 125 Summer Documentation. Web. 05 Apr Weisstein, Eric W. "Traveling Salesman Problem." From MathWorld--A Wolfram Web Resource Applegate, David L. The Traveling Salesman Problem: A Computational Study. Princeton: Princeton UP, Print. 7. Punnen, Abraham P. "The Traveling Salesman Problem: Applications, Formulations and Variations." Combinatorial Optimization The Traveling Salesman Problem and Its Variations (2007): Web. 8. "Expanding Neighborhood GRASP for the Traveling Salesman Problem." - Springer. Web. 05 Apr "Chapter 1." Traveling Salesman Problem: An Overview of Applications, Formulations, and Solution Approaches. Web. 05 Apr