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 Informatics, Edinburgh, UK Current sponsor: The Netherlands Organization for Scientific Research (NWO) Jano van Hemert — EvoCOP 2005 – p.1
I V N E U R S Motivation E I H T Y T O H F G R E U D I B N ① Combining studies on creating a TSP problem generator (van Hemert and Urquhart, 2004) Jano van Hemert — EvoCOP 2005 – p.2
I V N E U R S Motivation E I H T Y T O H F G R E U D I B N 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
I V N E U R S Motivation E I H T Y T O H F G R E U D I B N 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
I V N E U R TSP in one page S E I H T Y T O H F G R E U D I B N ✎ 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
I V N E U R TSP in one page S E I H T Y T O H F G R E U D I B N ✎ Objective (optimisation variant): in a graph, find the Hamiltonian cycle with minimal length ✎ Restriction: symmetric instances, thus distance(x,y) = distance(y,x) 400 350 300 250 200 150 100 50 0 0 50 100 150 200 250 300 350 400 Jano van Hemert — EvoCOP 2005 – p.3
I V N E U R TSP in one page S E I H T Y T O H F G R E U D I B N ✎ Objective (optimisation variant): in a graph, find the Hamiltonian cycle with minimal length ✎ Restriction: symmetric instances, thus distance(x,y) = distance(y,x) 400 350 300 250 200 150 100 50 0 0 50 100 150 200 250 300 350 400 Jano van Hemert — EvoCOP 2005 – p.3
I V N E U R S E I H T Y T O H F G R E U D I B N Evolving hard to solve TSP instances Jano van Hemert — EvoCOP 2005 – p.4
I V N E U R S Main idea E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Main idea E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Main idea E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Main idea E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Main idea E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R The evolutionary process S E I H T Y T O H F G R E U D I B N Read initial Start population Run the TSP solver on each problem instance Save the more difficult Yes More difficult problem instance to problem found? the improved series No Use 2-tournament selection to create a pairs of parents Stop Crossover & mutation to create Yes new problem instances from the set of pairs of parents Maximum No Replace current population generations with offspring using elitism reached? Jano van Hemert — EvoCOP 2005 – p.6
I V N E U R S Two TSP solvers E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Two TSP solvers E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Two TSP solvers E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S E I H T Y T O H F G R E U D I B N Experiments Jano van Hemert — EvoCOP 2005 – p.9
I V N E U R Experiment description S E I H T Y T O H F G R E U D I B N ✔ 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
I V N E U R S Fitness E I H T Y T O H F G R E U D I B N 2.2e+06 2e+06 1.8e+06 Search effort of CLK 1.6e+06 1.4e+06 1.2e+06 1e+06 800000 600000 400000 200000 0 50 100 150 200 250 300 350 400 Number of generations Jano van Hemert — EvoCOP 2005 – p.11
I V N E U R Evolutionary process in action S E I H T Y T O H F G R E U D I B N best000.tsp best001.tsp best002.tsp best003.tsp best004.tsp 400 400 400 400 400 350 350 350 350 350 300 300 300 300 300 250 250 250 250 250 200 200 200 200 200 150 150 150 150 150 100 100 100 100 100 50 50 50 50 50 0 0 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 best005.tsp best006.tsp best007.tsp best008.tsp best009.tsp 400 400 400 400 400 350 350 350 350 350 300 300 300 300 300 250 250 250 250 250 200 200 200 200 200 150 150 150 150 150 100 100 100 100 100 50 50 50 50 50 0 0 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 best010.tsp best011.tsp best012.tsp best013.tsp best014.tsp 400 400 400 400 400 350 350 350 350 350 300 300 300 300 300 250 250 250 250 250 200 200 200 200 200 150 150 150 150 150 100 100 100 100 100 50 50 50 50 50 0 0 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Jano van Hemert — EvoCOP 2005 – p.12
I V N E U R S Results E I H T Y T O H F G R E U D I B N 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
I V N E U R Increase in difficulty S E I H T Y T O H F G R E U D I B N Mean LK−CC on Near outliers, <= 3.0 IQR Random set Far outliers, > 3.0 IQR CLK on Random set 0 80000 160000 240000 320000 Lin−Kernighan search effort LK−CC: Evolved set CLK: Evolved set CLK on TSP generator 0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 Lin−Kernighan search effort Jano van Hemert — EvoCOP 2005 – p.14
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