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I V N E U R S E I H T Y T O H F G R E U D I B N Property analysis of symmetric travelling salesman problem instances acquired through evolution Jano van Hemert < jvanheme@inf.ed.ac.uk > Current host: School of

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T H E U N I V E R S I T Y O F E D I N B U R G H

Property analysis of symmetric travelling salesman problem instances acquired through evolution

Jano van Hemert

<jvanheme@inf.ed.ac.uk> Current host: School of Informatics, Edinburgh, UK Current sponsor: The Netherlands Organization for Scientific Research (NWO)

Jano van Hemert — EvoCOP 2005 – p.1

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T H E U N I V E R S I T Y O F E D I N B U R G H

Motivation

① Combining studies on creating a TSP problem generator (van Hemert and Urquhart, 2004)

Jano van Hemert — EvoCOP 2005 – p.2

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T H E U N I V E R S I T Y O F E D I N B U R G H

Motivation

① Combining studies on creating a TSP problem generator (van Hemert and Urquhart, 2004) ② and studies on evolving binary constraint satisfaction problems (van Hemert, 2003)

Jano van Hemert — EvoCOP 2005 – p.2

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T H E U N I V E R S I T Y O F E D I N B U R G H

Motivation

① Combining studies on creating a TSP problem generator (van Hemert and Urquhart, 2004) ② and studies on evolving binary constraint satisfaction problems (van Hemert, 2003) ③ into a search for interesting structural properties of TSP instances.

Jano van Hemert — EvoCOP 2005 – p.2

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T H E U N I V E R S I T Y O F E D I N B U R G H

TSP in one page

✎ Objective (optimisation variant): in a graph, find the Hamiltonian cycle with minimal length ✎ Restriction: symmetric instances, thus distance(x,y) = distance(y,x)

Jano van Hemert — EvoCOP 2005 – p.3

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T H E U N I V E R S I T Y O F E D I N B U R G H

TSP in one page

✎ Objective (optimisation variant): in a graph, find the Hamiltonian cycle with minimal length ✎ Restriction: symmetric instances, thus distance(x,y) = distance(y,x)

50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400

Jano van Hemert — EvoCOP 2005 – p.3

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T H E U N I V E R S I T Y O F E D I N B U R G H

TSP in one page

✎ Objective (optimisation variant): in a graph, find the Hamiltonian cycle with minimal length ✎ Restriction: symmetric instances, thus distance(x,y) = distance(y,x)

50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400

Jano van Hemert — EvoCOP 2005 – p.3

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T H E U N I V E R S I T Y O F E D I N B U R G H

Evolving hard to solve TSP instances

Jano van Hemert — EvoCOP 2005 – p.4

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T H E U N I V E R S I T Y O F E D I N B U R G H

Main idea

✔ Represent TSP problem instances as a list of (x,y)-coordinates representing 100 cities on a map of size 400 × 400

Jano van Hemert — EvoCOP 2005 – p.5

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T H E U N I V E R S I T Y O F E D I N B U R G H

Main idea

✔ Represent TSP problem instances as a list of (x,y)-coordinates representing 100 cities on a map of size 400 × 400 ✔ Initial population comprises of a set of uniform randomly generated instances

Jano van Hemert — EvoCOP 2005 – p.5

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T H E U N I V E R S I T Y O F E D I N B U R G H

Main idea

✔ Represent TSP problem instances as a list of (x,y)-coordinates representing 100 cities on a map of size 400 × 400 ✔ Initial population comprises of a set of uniform randomly generated instances ✔ Create new instances using uniform crossover and mutation

Jano van Hemert — EvoCOP 2005 – p.5

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T H E U N I V E R S I T Y O F E D I N B U R G H

Main idea

✔ Represent TSP problem instances as a list of (x,y)-coordinates representing 100 cities on a map of size 400 × 400 ✔ Initial population comprises of a set of uniform randomly generated instances ✔ Create new instances using uniform crossover and mutation ✔ Maximise the search effort required by a TSP solver

Jano van Hemert — EvoCOP 2005 – p.5

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T H E U N I V E R S I T Y O F E D I N B U R G H

Main idea

✔ Represent TSP problem instances as a list of (x,y)-coordinates representing 100 cities on a map of size 400 × 400 ✔ Initial population comprises of a set of uniform randomly generated instances ✔ Create new instances using uniform crossover and mutation ✔ Maximise the search effort required by a TSP solver ✔ Use a generational scheme with elitism

Jano van Hemert — EvoCOP 2005 – p.5

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T H E U N I V E R S I T Y O F E D I N B U R G H

The evolutionary process

Start Read initial population Run the TSP solver

n each problem instance

Use 2-tournament selection to create a pairs of parents Crossover & mutation to create new problem instances from the set of pairs of parents Replace current population with offspring using elitism More difficult problem found? Save the more difficult problem instance to the improved series Maximum generations reached? Yes No Stop Yes No

Jano van Hemert — EvoCOP 2005 – p.6

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T H E U N I V E R S I T Y O F E D I N B U R G H

Two TSP solvers

✔ Two TSP solvers will be tested independently ✔ Both are improved versions of the famous Lin-Kernighan approximation algorithm (Lin and Kernighan, 1973) ✔ Lin-Kernighan core operation: edge exchanges in a tour ✔ The number of these exchanges is used to measure the search effort in time complexity

Jano van Hemert — EvoCOP 2005 – p.7

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T H E U N I V E R S I T Y O F E D I N B U R G H

Two TSP solvers

✔ Chained Lin-Kernighan (CLK) (Applegate et al., 2000) Aim: more robustness in the resulting tour Method: chaining of multiple runs of Lin-Kernighan

Jano van Hemert — EvoCOP 2005 – p.8

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T H E U N I V E R S I T Y O F E D I N B U R G H

Two TSP solvers

✔ Chained Lin-Kernighan (CLK) (Applegate et al., 2000) Aim: more robustness in the resulting tour Method: chaining of multiple runs of Lin-Kernighan ✔ Lin-Kernighan with Cluster Compensation (LK-CC) (Neto, 1999) Aim: reduction of computational effort, while maintaining quality of the tour Method: calculate cluster distance for nodes, used to guide the utility function of Lin-Kernighan

Jano van Hemert — EvoCOP 2005 – p.8

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T H E U N I V E R S I T Y O F E D I N B U R G H

Experiments

Jano van Hemert — EvoCOP 2005 – p.9

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T H E U N I V E R S I T Y O F E D I N B U R G H

Experiment description

✔ 190 independent runs of the evolutionary algorithm ✔ One run comprises of 600 generations ✔ The population size is set to 30 ✔ Storing each run’s overall best solution, i.e., hardest to solve problem instance ✔ The set of best solutions is called: Algorithm:Evolved set (e.g., CLK:Evolved set) ✔ Also, a Random set exists with (190×30=) 5 700 instances generated uniform randomly

Jano van Hemert — EvoCOP 2005 – p.10

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T H E U N I V E R S I T Y O F E D I N B U R G H

Fitness

200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 1.6e+06 1.8e+06 2e+06 2.2e+06 50 100 150 200 250 300 350 400 Search effort of CLK Number of generations

Jano van Hemert — EvoCOP 2005 – p.11

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T H E U N I V E R S I T Y O F E D I N B U R G H

Evolutionary process in action

50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best000.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best001.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best002.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best003.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best004.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best005.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best006.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best007.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best008.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best009.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best010.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best011.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best012.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best013.tsp 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 best014.tsp

Jano van Hemert — EvoCOP 2005 – p.12

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T H E U N I V E R S I T Y O F E D I N B U R G H

Results

① Increase in difficulty after evolving ② Clustering properties of instances ③ Distribution of segment lengths in instances ④ Distribution of pair-wise distances in instances ⑤ Swapping evolved sets for the two solvers

Jano van Hemert — EvoCOP 2005 – p.13

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T H E U N I V E R S I T Y O F E D I N B U R G H

Increase in difficulty

Lin−Kernighan search effort

80000 160000 240000 320000 LK−CC on Random set CLK on Random set Mean Near outliers, <= 3.0 IQR Far outliers, > 3.0 IQR

LK−CC: Evolved set CLK: Evolved set CLK on TSP generator

Lin−Kernighan search effort

1e+06 2e+06 3e+06 4e+06 5e+06 6e+06

Jano van Hemert — EvoCOP 2005 – p.14

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T H E U N I V E R S I T Y O F E D I N B U R G H

Clustering properties

✔ Use the clustering algorithm GDBSCAN (Sander et al., 1998) to determine the amount of clusters in TSP instances ✔ Problem with clustering algorithms: results depend highly on input parameters ✔ Work-around: sweep the parameter space and show the cumulative results

Jano van Hemert — EvoCOP 2005 – p.15

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T H E U N I V E R S I T Y O F E D I N B U R G H

Clustering properties

① For every TSP problem instance in the set: ② Run GDBSCAN within a fixed and sensible range of its parameters ③ For every parameter, record how many clusters are found ④ Count for every number of clusters the number of

ccurrences

Jano van Hemert — EvoCOP 2005 – p.16

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T H E U N I V E R S I T Y O F E D I N B U R G H

Clustering properties

10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 Average occurrences of clusterings Number of clusters Random set CLK:Evolved set LK-CC:Evolved set

Jano van Hemert — EvoCOP 2005 – p.17

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T H E U N I V E R S I T Y O F E D I N B U R G H

Distribution of segment lengths

① Given a TSP, create a tour using a TSP solver ② Calculate the length of all the segments of that tour ③ Repeat this for every TSP instance in the set ④ Create histograms over the averaged results

Jano van Hemert — EvoCOP 2005 – p.18

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T H E U N I V E R S I T Y O F E D I N B U R G H

Distribution of segment lengths

5 10 15 20 25 30 10 20 30 40 50 60 Average occurrences Segment lengths CLK:Random set 5 10 15 20 25 30 10 20 30 40 50 60 Average occurrences Segment lengths CLK:Evolved set

Jano van Hemert — EvoCOP 2005 – p.19

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T H E U N I V E R S I T Y O F E D I N B U R G H

Distribution of segment lengths

5 10 15 20 25 30 20 40 60 80 100 120 140 160 180 Average occurrences Segment lengths CLK, Random set CLK:Evolved set LK-CC, Random set LK-CC:Evolved set

Jano van Hemert — EvoCOP 2005 – p.20

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T H E U N I V E R S I T Y O F E D I N B U R G H

Distribution of pair-wise distances

50 100 150 200 250 300 350 400 100 200 300 400 500 600 Average occurrences Pair-wise distances Random set CLK:Evolved set LK-CC:Evolved set

Jano van Hemert — EvoCOP 2005 – p.21

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T H E U N I V E R S I T Y O F E D I N B U R G H

Swapping evolved sets

CLK CC-LK CLK:Evolved set

1 753 790 (251 239) 207 822 (155 533)

CC-LK:Evolved set

268 544 (71 796) 1 934 790 (799 544) Random set 130 539 (34 452) 19 660 (12 944)

✎ Mean and standard deviation, in brackets, of the search effort required by both algorithms on the Random set and both Evolved sets

Jano van Hemert — EvoCOP 2005 – p.22

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T H E U N I V E R S I T Y O F E D I N B U R G H

Swapping evolved sets

CLK CC-LK CLK:Evolved set

1 753 790 (251 239) 207 822 (155 533)

CC-LK:Evolved set

268 544 (71 796) 1 934 790 (799 544) Random set 130 539 (34 452) 19 660 (12 944)

✎ Mean and standard deviation, in brackets, of the search effort required by both algorithms on the Random set and both Evolved sets

Jano van Hemert — EvoCOP 2005 – p.22

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T H E U N I V E R S I T Y O F E D I N B U R G H

Swapping evolved sets

CLK CC-LK CLK:Evolved set

1 753 790 (251 239) 207 822 (155 533)

CC-LK:Evolved set

268 544 (71 796) 1 934 790 (799 544) Random set 130 539 (34 452) 19 660 (12 944)

✎ Mean and standard deviation, in brackets, of the search effort required by both algorithms on the Random set and both Evolved sets

Jano van Hemert — EvoCOP 2005 – p.22

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T H E U N I V E R S I T Y O F E D I N B U R G H

Conclusions

✔ Using an evolutionary algorithm we can find TSP problems that are very difficult to solve for a particular

TSP solver

Jano van Hemert — EvoCOP 2005 – p.23

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T H E U N I V E R S I T Y O F E D I N B U R G H

Conclusions

✔ Using an evolutionary algorithm we can find TSP problems that are very difficult to solve for a particular

TSP solver

✔ No order parameter that predicts difficulty in general was found

Jano van Hemert — EvoCOP 2005 – p.23

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T H E U N I V E R S I T Y O F E D I N B U R G H

Conclusions

✔ Using an evolutionary algorithm we can find TSP problems that are very difficult to solve for a particular

TSP solver

✔ No order parameter that predicts difficulty in general was found ✔ The layout of cities in a TSP highly influences the difficulty,

CLK: clustering plays an integral role LK-CC: problem with robustness

to slight perturbations in layouts

Jano van Hemert — EvoCOP 2005 – p.23

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T H E U N I V E R S I T Y O F E D I N B U R G H

Conclusions

✔ Using an evolutionary algorithm we can find TSP problems that are very difficult to solve for a particular

TSP solver

✔ No order parameter that predicts difficulty in general was found ✔ The layout of cities in a TSP highly influences the difficulty,

CLK: clustering plays an integral role LK-CC: problem with robustness

to slight perturbations in layouts ✔ A generic technique, but tailoring is required to the landscape of problem instances in combination with a solver

Jano van Hemert — EvoCOP 2005 – p.23

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T H E U N I V E R S I T Y O F E D I N B U R G H

Future work

✔ Test more TSP solvers

Jano van Hemert — EvoCOP 2005 – p.24

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T H E U N I V E R S I T Y O F E D I N B U R G H

Future work

✔ Test more TSP solvers ✔ Create instances against ensembles of solvers

Jano van Hemert — EvoCOP 2005 – p.24

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T H E U N I V E R S I T Y O F E D I N B U R G H

Future work

✔ Test more TSP solvers ✔ Create instances against ensembles of solvers ✔ Define more structural properties that may help explain the difficulty of problem instances, specifically for TSP

Jano van Hemert — EvoCOP 2005 – p.24

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T H E U N I V E R S I T Y O F E D I N B U R G H

Future work

✔ Test more TSP solvers ✔ Create instances against ensembles of solvers ✔ Define more structural properties that may help explain the difficulty of problem instances, specifically for TSP ✔ Use the general process of evolving (difficult) problem instances in other areas (suggestions welcome!)

Jano van Hemert — EvoCOP 2005 – p.24

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T H E U N I V E R S I T Y O F E D I N B U R G H

Thanks for your attention

✎ Slides and publications are available from http://google.com/search?q=jano+tsp

Jano van Hemert — EvoCOP 2005 – p.25

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References

Applegate, D., Cook, W., and Rohe, A. (2000). Chained lin-kernighan for large travelling salesman problems. http://www.citeseer.com/applegate99chain Lin, S. and Kernighan, B. (1973). An effective heuristic algorithm for the traveling salesman problem. Oper- ations Research, 21:498–516. Neto, D. (1999). Efficient Cluster Compensation for Lin- Kernighan Heuristics. PhD thesis, Computer Sci- ence, University of Toronto. Sander, J., Ester, M., Kriegel, H.-P., and Xu, X. (1998). Density-based clustering in spatial databases: The algorithm GDBSCAN and its applications. Data Min.

Knowl. Discov., 2(2):169–194.

van Hemert, J. (2003). Evolving binary constraint satisfaction problem instances that are difficult to solve. In Proceedings of the IEEE 2003 Congress on Evolu- tionary Computation, pages 1267–1273. IEEE Press. van Hemert, J. (2005). Property analysis of symmetric travelling salesman problem instances acquired through evolution. In Raidl, G. and Gottlieb, J., editors, Evolutionary Computation in Combinatorial 25-1

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Optimization, Springer Lecture Notes on Computer Science, pages 122–131. Springer-Verlag, Berlin. van Hemert, J. and Urquhart, N. (2004). Phase transi- tion properties of clustered travelling salesman problem instances generated with evolutionary computa-

tion. In Yao, X., Burke, E., Lozano, J. A., Smith,

J., Merelo-Guerv´

s, J. J., Bullinaria, J. A., Rowe, J.,

Kab´ an, P. T. A., and Schwefel, H.-P., editors, Parallel Problem Solving from Nature (PPSN VIII), volume 3242 of LNCS, pages 150–159, Birmingham, UK. Springer-Verlag. 25-1