State Space Reduction for the SYMMETRIC Traveling Salesman Problem through Halves Tours Complement

Transcription

1 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e 64 State Space Reduction for the SYMMTRI Traveling Salesman Problem through Halves Tours omplement Kamal R. l-rawi epartment of omputer Science, Faculty of Information Technology, Petra University, mman, JORN. kamalr@uop.edu.jo k_alrawi@yahoo.com bstract: The Traveling Salesman Problem (TSP) is an NP-hard. The state space increases exponentially with the number of nodes N. The number of paths is (N-1)!. There are two main points in the TSP we took advantage of: The number of states N in a path is known in advance, which is the number of nodes in the tour; and touring each node only once. The complete tour can be constructed by concatenating a half path with its complements. This limit a partial path no more than its half way to the complete tour. Limiting the partial path just to the half way of the complete tour reduces sharply the state space and the searched state space as well. This leads to reduction in both memory requirement and execution time which are the major challenges for computer scientist to tackle the TSP with exact algorithms. Keywords: Traveling Salesman Problem, TSP, state space reduction, state space search, optimal path, Half Tour omplement for TSP, HT-TSP. 1. Introduction: The TSP represents N nodes that we have to find the optimal tour that starts at a node, visits every other node exactly once, and returns to the starting node. The TSP can be considered as a graph G with vertex set V and edge set : G(V,); V={1,,N};={cost c ij ; i=1,,n-1;j=1, N} For asymmetric graph c ij c ji, however for symmetric graph, which is our concern here, c ij =c ji. The TSP is an NP-hard. s the number of nodes increase the number of paths increases exponentially. For N nodes, the total number of paths are (N-1)!. Optimal solutions to small instances can be found in reasonable time; however, it will be very time consuming to solve large instances with optimal algorithms.. The TSP has many applications in the real words: areas of vehicle routing, workshop scheduling and computer wiring (Lawler et al. 1985), logistics, genetics, manufacturing and telecommunications (pplegate et al. 2007). There are many exact algorithms for the TSP in the literature. However, since it is an NP-hard problem many approximation algorithms have been developed. Many local search algorithms have been published. Such algorithms are needed when an acceptable (we mean non optimal nor even near optimal) path is required due to lack of time, since finding the optimal path is time consuming task. TSP has been conducted by many authors with different approximation algorithms: divide and conquer (Valenzuela and Jones1995) greedy algorithm (ryant 1995), genetic algorithms (hatterjee and et al 1996, arter and Ragsdale 2006, Snyder and. askin 2006), the antzig-fulkerson-johnson algorithm (pplegate et al 2004), heuristic lgorithms (Valenzuela and Williams 1997, Tsai et al 2004), nt colonies (origo and Gambardella 1997), neural network (Takahashi et al 2002). 2. Objective: This work has been conducted to improve the performance of the optimal algorithms to find the optimal tour for the symmetric TSP. Performance improvement comes from reducing the state space,

2 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e 65 and reducing the searched state space. This leads to reduction in both memory requirement and the execution time. 3. The algorithm development: ranch-and-ound (-n-) is an exact algorithm that is typically used to find optimal solutions of optimization problems. However, employing it for the TSP for large N is very tedious task. We will try in this work to reduce the state space and searched state space to obtain the optimal path for the TSP using -n- algorithm. 3.1 Ordinary -n- optimal search for TSP It is well known that number of paths for the TSP with N nodes is (N-1)!. The number of states in the state space is: 1+(N-1)+(N-1)*(N-2)+ (N-1)*(N-2)*(N-3)+ + (N-1)* *4*3+3(N-1)*(N-2)* *2 We can rewrite the above formula as: N 1 S N = t0 + [ ti 1( N i)] + t N 2; to = 1; and t N 1 = t N 2 = N i= omplement approach (this work) There are two main points in the TSP we have to take advantage of: The number of states N in the optimal path is known in advance, which is the number of nodes in the tour; and touring each node only once. This let us branch a path no more than its half way to the complete tour. The complete tour can be constructed by concatenating halves partial paths with their complements halves partial paths. In order to explain it we have to give an example. Let us say we have five cities to tour. The complement halves partial paths for the half partial path are and. These halves partial paths are complement for the half partial path too. So, when we have the partial paths or and the partial paths of their complements or then a complete tour can be constructed. For seven cities FG, the complement halves partial paths for the half partial path are FG, GF, FG, FG, GF, and GF. These halves partial paths are complement for halves partial paths,,,, and. So, a complete tour can be constructed by concatenating a partial path from (,,,,, ) with a partial path from their complements (FG, GF, FG, FG, GF, GF). Limiting the partial path branching just to the half way of the complete tour reduces the state space of the TSP and decreases sharply the searched state space. This leads to reduction in both memory requirement and execution time which are the major challenges for computer scientist to tackle the TSP for optimal path. The number of states in the state space is: 1+(N-1)+(N-1)*(N-2)+ (N-1)*(N-2)*(N-3)+ + (N-1)*(N-2)* *(N-N/2) We can rewrite the above formula as: N / 2 S N = t0 + [ ti 1( N i)] + t N 2; to = 1; and t N 1 = t N 2 = N i= 1 The reduction in the state space is less than the ordinary approach by: (N-1)* *(N-N/2-1)+ (N-1)* *(N-N/2-2)+ +3(N-1)* *2 This is a very large reduction for the state space. This reduction increases as N increases. Figure-1 shows the percentage of the state space using complement approach relative to the ordinary approach. It decreases exponentially with number of nodes N. For a problem with 20 nodes we reduced the state!!

3 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e 66 space down to %. The searched state space is reduced too since the optimal tour is constructed by combining the optimal half path with its complements. complement/ordinary state space% number of cities N in the tour Figure1: complemented to ordinary percentage of the state space. It reduced exponentially with the number of nodes N. This reduces sharply the state space Table1:ost matrix chosen randomly between 1 and 20

4 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e Figure2: State space for data of table-1. The state space inside the heavy dotted line represents the State space for Half Tour omplement (HT). It is much less than the whole state space. Heavy and very heavy solid represent the search for normal -n- and HT, respectively. Figure-2 shows the whole nodes represent the whole state space for given data in table-1. The nodes that are located inside the heavy dotted circle represent the state space for Half Tour omplement (HT) approach. The searched state space for normal -n- (heavy and very heavy solid nodes) is /89*100=46.4% of the whole state space. While for HT (very heavy solid nodes) is 11 /89*100=.4%. The reduction for HT relative to normal -n-is 374%. The complete tour is. It is constructed from the half tour and its complement. 3.3 State space search using HT approach for the given data. OPN[(0)] HalfTour[ ]

5 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e 68 OPN[(1), (5), (10), (11)] HalfTour[ ] OPN[ (5), (10), (11)] HalfTour[(6),( 10), (15)] OPN[ (5), (10), (11)] HalfTour[(6), (9), ( 10), (), (15), (20)] We have found the half tour (9) and its complement (15). The final complete tour is +=9+15+2( cost)=. See the appendix for state space search using normal -n- approach. 4. onclusion: The complete path is constructed by concatenating the optimal half partial path with its optimal half partial path. This limit a partial path no more than its half way to the complete tour. This leads to: 1- reducing the state space, 2- reducing the searched state space., and 3- reducing number of paths in the state space. This leads to reduction in memory requirement and the execution time. So, we can conduct an optimal path for a larger value of number of nodes N with the same machines and time constrain that we have. 5. References: pplegate,. L., ixby, R.., hvatal, V., and ooke, W. J., (2007), "The travel salesman problem", Princeton Univ. Press. pplegate., ixby R., hvatal V., ook W. "Implementing the antzig-fulkerson-johnson algorithm for large traveling salesman problems", Mathematical Programming 97 (2004) ryant. J.. "Very greedy crossover in a genetic algorithm for the traveling salesman problem.", In K. M. George, Janice H. arroll, d eaton, ave Oppenheim, and Jim Hightower, editors, Proceedings of the 10th M Symposium on pplied omputing, 4 8, M Press, New York. arter.., and Ragsdale.T. " new approach to solving the multiple traveling salesperson problem using genetic algorithms", uropean J. of Operational Research 246 7, hatterjee S., arrera., and Lynch L. "Genetic algorithms and traveling salesman problems.", uropean Journal of Operational Research, 93(3): , 20, origo M., Gambardella L.M. "nt colonies for the traveling salesman problem", io Systems, 73 81, Lawler,. L.; Lenstra, J. K.; Kan,. H. G. R.; and Shmoys,.. (1985). The Traveling Salesman Problem. ssex, ngland: John Wiley & Sons. Sahni, S., (1976),"General techniques for combinational approximations," Oper. Res., Vol., pp: Snyder L. V., askin M.S. " random-key genetic algorithm for the generalized traveling salesman problem", uropean J. of Operational Research 174 (1) 38 53, Takahashi S., Fujimura K., Tokutaka H. The SOM-TSP method for the three-dimension city location problem, Neural Information Processing, IONIP 02, Proceedings of the 9th International onference 5 (2002) Tsai.F., Tsai.W., Tseng.. " new hybrid heuristic approach for solving large traveling salesman problem", Information Sciences 166 (1 4) 67 81,2004. Valenzuela L and Jones. J., " parallel implementation of evolutionary divide and conquer for the TSP.", In Proceedings of the First I/I International onference on Genetic lgorithms in ngineering Systems: Innovations and pplications, , Sheffield (UK),.-14, Valenzuela. L., and Williams L. P. "Improving Simple Heuristic lgorithms for the Travelling Salesman Problem using a Genetic lgoritm", Proc. of the Seventh Int.onf. on Genetic lgorithms, pp , 1997.

6 T h e R e s e a r c h u l l e t i n o f J o r d a n M, V o l u m e I I ( I I ) P a g e 69 ppendix: Search the state space using normal ranch-n-ound for the given example. OPN[(0)] OPN[(1), (5), (10), (11)] OPN[ (5), (6), ( 10), (10), (11) (15)] OPN[ (6), (9), ( 10), (10), (11), (), (15), (20)] OPN[(9), ( 10), (10), (11), (), (), (15), (20), (23)] OPN[ ( 10), (10), (11), (11), (), (), (15), (18), (20), (23)] OPN[ (11), (11), (), ( ), (), (), (14), (14), (15), (18), (19), (20), (23)] OPN[ (11), (), (), (), (), (14), (14), (15), (18), (19), (20), (23), ()] OPN[ (), (), (), (), (13), (14), (14), (15), (18), (19), (20), (23), (), (), ()] OPN[ (), (), (), (13), (14), (14), (14), (15), (18), (19), (20), (23), (), (), ()] OPN[ (), (), (13), (14), (14), (14), (15), (18), (19), (20), (23), (), (), (), (27)] OPN[ (), (13), (14), (14), (14), (15), (18), (19), (20), (21), (23), (), (), (), (), (27)] OPN[ (13), (14), (14), (14), (15), (18), (19), (20), (21), (23), (), (), (), (), (), (27), (27)] OPN[ (14), (14), (14), (15), (17), (18), (19), (20), (21), (22), (23), (), (), (), (), (), (27), (27)] OPN[ (14), (14), (15), (17), (18), (19), (20), (21), (22), (23), (), (), (), (), (), (), (27), (27)] is a complete tour.