Hierarchical Solution of the Traveling Salesman Problem with Random Dyadic Tilings

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1 Hierarchical Solution of the Traveling Salesman Problem with Random Dyadic Tilings Tamás Kalmár-Nagy and Bendegúz Dezső Bak Department of Fluid Mechanics, Faculty of Mechanical Engineering Budapest University of Technology and Economics Abstract We propose a hierarchical heuristic approach for solving the Traveling Salesman Problem (TSP) in the unit square. The points are partitioned with a random dyadic tiling and clusters are formed by the points located in the same tile. Each cluster is represented by its geometrical barycenter and a coarse TSP solution is calculated for these barycenters. Midpoints are placed at the middle of each edge in the coarse solution. Near-optimal (or optimal) minimum tours are computed for each cluster. The tours are concatenated using the midpoints yielding a solution for the original TSP. The method is tested on random TSPs independent, identically distributed points in the unit square) up to points as well as on a popular benchmark problem (att532 - coordinates of 532 American cities). Our solutions are 8-13% longer than the optimal ones. We also present an optimization algorithm for the partitioning to improve our solutions. This algorithm further reduces the solution errors (by several percent using 1000 iteration steps). The numerical experiments demonstrate the viability of the approach. 1 Introduction The Traveling Salesman Problem (TSP) has generated a tremendous amount of research. It is an easy to state, yet hard to solve combinatorial problem: given n points, find the minimum length tour connecting all points that starts and ends at the same point. The TSP is an NP-hard problem [1]. Thorough studies of the TSP are given by Laporte [2] and Applegate et. al [3], the latter includes vast amount of computational results. Another overview of exact and approximate methods is given by Matai et al. [4], as well as examples of possible applications including vehicle routing and drilling of printed circuit boards. The most popular exact methods for solving the TSP are the branch-and-bound methods [5, 6] and the branch-and-cut method [3]. The main drawback of these (and other) exact methods is the extensive amount of computer time they require to find the optimal solution of large TSP instances (10000 points or more). Approximate methods were developed to provide high quality solutions [7] quickly. These heuristics are either tour construction or tour improvement type methods. Tour construction methods gradually build up the TSP solution. Approximate methods which rely on the use of spacefilling curves [8, 9, 10] and partitioning algorithms [11, 12, 13] are tour construction methods. Tour improvement methods generate an initial tour and try to alter this tour to become shorter. Tour improvement using genetic algorithm [14, 15, 16, 17] is a popular approximate method. Another tour improvement type method is the 2-opt method [18, 19] which reorders the TSP solution to reduce tour length. It takes a section from the tour and adds it in reverse order to form a new tour as shown in Figure 1. dyadic@kalmarnagy.com bak@ara.bme.hu 1

2 (a) (b) Figure 1: The part of the solution (a) before and (b) after the application of the 2-opt method. In this example there are crossing edges in the original solution (Figure 1(a)). Points 4 and 5 are swapped (added in reverse order to the original solution), thus the crossing edges are removed and the solution becomes shorter (Figure 1(b)). The 2-opt method is often incorporated into other heuristics for local optimization. The generalization of 2-opt leads to the very fast Lin-Kernighan heuristic [20, 21, 22] which is currently considered one of the best approximate methods for the TSP. The major drawback of approximate methods is that they often get stuck at a local minimum. This can be prevented using a metaheuristic such as ant colony optimization [23] or simulated annealing [24, 25, 26]. An overview of these metaheuristics is given by Johnson et al. [27]. Focusing on partitioning algorithms, an influential piece of work by Karp [11] considers such algorithms for the approximate solution of large instances of the TSP. To reduce the complexity of the problem, partitioning algorithms subdivide the set of points into small groups, construct an optimum tour through each group, and then patch these tours together to form a closed tour through all the points. Figure 2 shows the basic idea. (a) (b) (c) (d) Figure 2: (a) A set of n points, (b) is first clustered into disjoint sets. (c) A subsolution is obtained for each cluster, (d) and these are concatenated. First the starting set of points (depicted in Figure 2(a)) is reduced : the points are grouped into disjoint subsets (clusters) as shown in Figure 2(b). Good quality subsolutions are computed for each cluster of points (Figure 2(c)). A cluster of points is replaced with a representative point, and the subsolutions are concatenated (Figure 2(d)). Yoshiyuki and Yoshiki [12] introduced a method in which the unit square was recursively subdivided into smaller squares (regular square tiling). After each division the points located in the same subsquare were substituted with the barycenter of that subsquare. For every subdivision an approximate solution was constructed for the barycenters, based on the solution in the previous step. The work of Ugajin [13] relies on Yoshiyuki s method, including a moving-frame renormalization scheme. In this approach the tiles are generated based on an intensity function, which has peaks around clusters of points. Xiang et al. [28] proposed a method to partition the points into four segments (groups) based on the coordinates of the points. Four corner points were designated in advance to serve as endpoints for the tours computed in each segment. Another hierarchical approach is proposed by Houdayer et al. [15]. They combined renormalization with a genetic algorithm. 2

3 In this work, we propose a new partition-and-cluster approach. The set of points is partitioned with dyadic tilings [29, 30], yielding clusters of points. A TSP solution is calculated for the barycenters of the clusters. A tour with different endpoints (subsolution) is calculated for the points of each cluster. These tours are concatenated based on the TSP solution of the barycenters, yielding the global solution. The global solution strongly depends on the tiling, but this dependence is unknown a priori. A large number of TSPs consisting of n independent, identically distributed points in the unit square is solved using random dyadic tilings to investigate the solution quality. To improve the efficacy of the algorithm we introduce the Genetic Algorithm Enhanced Hierarchical Solution (GAEHS) method which borrows elements of the classical genetic algorithm [31, 32, 33]. The genetic operators randomly choose and modify parts of the tiling. The modified tiling is kept if it is better than its predecessor. This paper is structured as follows: in Section 2 we describe dyadic tiling, its connection with labeled complete binary trees and how the set of points is partitioned. In Section 3 we explain how a global solution of the TSP can be constructed in a hierarchical manner. In Section 4 results are presented for random TSPs and in Section 5 GAEHS is discussed and the improvements are demonstrated. 2 Tiles, trees and partitioning Tiling (or tessellation) is fully covering an area with smaller, non-overlapping plane figures. The partitioning of the plane to find objects intersecting a specified area is a common approach [34, 35]. Tiling has been applied to a range of applications, including image or audio processing [36] and meshing complex geometry [37]. 2.1 Dyadic tiling A dyadic rectangle (referred to as tile) in the unit square is defined as T (a, b, s, t) = [a 2 s, (a + 1) 2 s ] [b 2 t, (b + 1) 2 t ], (1) where the 4-tuple (a, b, s, t) of nonnegative integers satisfy 0 a < 2 s, 0 b < 2 t. (2) The order of a tile is l = t + s. The unit square itself is T (0, 0, 0, 0), the unique 0th order tile. An lth order dyadic tiling partitions the unit square into 2 l tiles of area 2 l [29, 30]. There is a correspondence between dyadic tilings and labeled complete binary trees (HV -trees) [30, 34]. Figure 3 illustrates such a correspondence. (a) (b) (c) Figure 3: (a) A 2nd order dyadic tiling, (b) illustration of the subdivision process, and (c) the corresponding HV -tree. Figure 3(b) illustrates the cutting process. The unit square is depicted on the top. The labels H and V indicate whether the tile above them was halved horizontally or vertically. The loose edges at the bottom correspond to the tiles shown next to them. These tiles constitute the tiling shown in Figure 3(a). If the 3

4 loose edges and the tiles are removed from this figure, we get a so-called HV -tree as shown in Figure 3(c). The HV -tree uniquely defines the tiling. The height (or order) of the HV -tree equals to the order of the dyadic tiling. For a given HV -tree the 4-tuple (a, b, s, t) can be calculated for each tile. If the 4-tuple of an lth order tile is (a, b, s, t), the 4-tuple of its left (L) and right (R) l + 1th order children tiles are calculated as HL(a, b, s, t) = (a, 2b, s, t + 1) HR(a, b, s, t) = (a, 2b + 1, s, t + 1) VL(a, b, s, t) = (2a, b, s + 1, t) VR(a, b, s, t) = (2a + 1, b, s + 1, t), where HL, HR, VL, VR are R 4 R 4 affine transformations. Although every HV -tree uniquely defines the corresponding dyadic tiling, the reverse is not true. The number of HV -trees which represent every distinct lth order dyadic tiling is given by the recurrence [30] The first few elements of the sequence are 2.2 Tessellation and partitioning N l = 2N 2 l 1 N 4 l 2, N 0 = 1, N 1 = 2. (4) N 2 = 7, N 3 = 82, N 4 = 11047, N 5 = ,.... (5) Let HV(l) denote the set of lth order HV -trees. A dyadic tiling T HV(l) provides a natural tessellation of the unit square. Given a set of points X in the unit square, let T (X) denote the set of clusters obtained by grouping points belonging to the same tile. This is a set theoretical partitioning, i.e. the elements of X are grouped into non-empty subsets such that every element is included in exactly one of the subsets. The size of a cluster is the number of points it contains. Figure 4 shows a dyadic tiling and the corresponding clusters (encircled). (3) Figure 4: The unit square is partitioned with a dyadic tiling. The clusters of points (dots) are encircled. The barycenters (squares) of the clusters are also depicted. The barycenters of the clusters are the centroids of the clusters, the set of barycenters is denoted by B. There is a one-to-one correspondence between barycenters and tiles; in other words, a barycenter uniquely determines its tile. Barycenters are clustered similarly to the original points. 4

5 3 Hierarchical method for solving the TSP 3.1 Solution with one partitioning Consider a set of points X in the unit square (gray dots in Figure 5(a)). Our method consists of the following steps to obtain a TSP solution: 1. X is partitioned with a tiling T which yields the set of clusters T (X) and the corresponding set of barycenters B. The barycenters are denoted by (squares) in Figure 5(a). 2. A coarse TSP solution (Figure 5(b)) is calculated for the barycenters. Midpoints are then placed at the middle of each edge. Two midpoints naturally belong to each barycenter (located on the edges emanating from the barycenter). 3. Each barycenter is replaced by the cluster of points it represents (these points are located in its tile as shown in Figure 5(c)). Thus two midpoints and a cluster of points belong to every barycenter. The (approximately) minimum tour between these two midpoints through the points is computed. The union of all such tours forms a closed tour. Figure 5(c) highlights each tour with alternating continuous and dashed lines. 4. Midpoints are removed from the tour to yield the TSP solution for the original points. At this point the 2-opt method is used to improve the final solution (Figure 5(d)). 5

6 (a) (b) (c) (d) Figure 5: (a) Partitioning the points, (b) coarse TSP solution for the barycenters and placement of the midpoints, (c) minimum tours are calculated between midpoints and (d) the final solution using 2-opt. The algorithm is simple and works for small number of points (n 100). For larger instances of TSP one must deal with the following issues: If only a few tiles are used for partitioning, there are too many points in each cluster. This makes tour constructions in Step 3 computationally slow. If the order of the tiling is high, there will be too many barycenters. In this case the construction of the optimal or a near-optimal coarse TSP solution becomes slow. That is, the method requires small number of clusters, while the average number of points in each cluster must also be small, i.e. both the number of clusters and the average number of points in each cluster must be orders of magnitude smaller than n. 3.2 Hierarchical solution with multiple partitionings To overcome the above issues with large TSPs, we propose a hierarchical version of our algorithm. At first, X is partitioned with a fine tiling T 1, so that the average cluster size is sufficiently small (preferably less than 6

7 10 points). This yields the set of clusters C 1 = T 1 (X) and the set of barycenters B 1. As discussed, in this case the number of barycenters will be too high (usually their number is in the same order of magnitude as n) to efficiently construct the coarse TSP solution. Instead, the barycenters are partitioned with another tiling T 2 which is coarser than T 1 (i.e. we obtain C 2 = T 2 (B 1 ) and B 2 ). The new barycenters are further partitioned with gradually coarser tilings T 3,..., T i, T i+1,..., T k. The number of barycenters in B k is assumed to be sufficiently small (2-3 orders of magnitude smaller than n) to construct the coarse TSP solution. Based on Steps 3 and 4, the TSP solution for any B i 1 is built from that of B i. The TSP solution for the original points is built from that of B 1. The set of tilings (T 1, T 2,..., T k ) used for the partitioning is called the tiling combination. 7

8 T 1 X Remove + use 2-opt keep + T 2 B 1 Remove 2-opt add + keep add keep Figure 6: Hierarchical solution of the TSP using two partitionings. Legend: are the points, are the barycenters of C 1 and are the barycenters of C 2, The midpoints are denoted by and +. In Figure 6 a solution is shown step-by-step using two tilings: T 1 and T 2. X is partitioned with T 1 which yields the set of barycenters B 1 (squares). B 1 is partitioned further with T 2 (which is coarser than T 1 ) yielding the set of barycenters B 2 (triangles). The coarse TSP solution is computed for B 2 and midpoints are placed. To every barycenter belong two midpoints and its barycenters from the first partitioning. The TSP solution for B 1 is built from that of B 2. The midpoints are removed during the process. New midpoints + are added to the middle of each edge in the TSP solution for B 1. Two midpoints + and a cluster of points belong to each barycenter. The TSP solution for the original points is built from that of B 1. The midpoints + are removed during the process. 8

9 4 Computational Results The hierarchical algorithm of Section 3 was tested for n independent, identically distributed points in the unit square. The tiling combinations were chosen so as to keep the average size of clusters around 2-4, i.e. 2 n/2 l 4, where l is the order of the first tiling. For instance, 500 points are first partitioned with a 7th order tiling, thus the average number of points in each cluster is 500/ The orders of the subsequent tilings in the tiling combination are l 2, l 4... (the orders of the subsequent tilings are 5 and 3 for 500 points). The subsequent partitionings reduce the number of points to be connected by 2 orders of magnitude. The Concorde software [3] is used to calculate optimal/near-optimal solutions. 4.1 Mean solution lengths and error distributions In a seminal paper Beardwood et al. [38] proved that the expected optimal tour length L min for n independent, identically distributed points in the unit square behaves as lim L min/ n = β, (6) n where β [39] is the Beardwood-Halton-Hammersley constant. Many studies dealt with the numerical approximation of β [39, 40, 41] or the determination of its lower and upper bounds [42]. The approximation L min = β n + C. (7) is consistent with Eq. (6). Using Table 16.7 from Chapter 16 of [3] yielded β = 0.71 and C = Figure 7 shows L min (solid line), and the mean lengths of our solutions (circles) for various n s. Solution length Mean solution length 0.71 n Numberofpointsn Figure 7: The L min = 0.71 n+0.63 curve and the mean tour lengths obtained with our hierarchical method. TSP solutions were computed for n = independent, identically distributed points in the unit square. The coarse TSP solution was always calculated exactly (by brute force or Concorde). Exact solutions were calculated for clusters containing up to 5 points, and near-minimum tours were calculated for larger clusters. The results togethether with the order of tilings are summarized in Table 1. The optimal/near-optimal TSP solutions were calculated using Concorde. The solution error shows how much longer is (on the average) the tour we found compared to the optimal/near-optimal one. 9

10 Table 1: Mean and standard deviation of the solution error distribution for random TSPs. Number of points No. of TSPs Order of tilings Mean error [%] Std of error [%] (5, 3) (6, 4, 2) (7, 5, 3) (8, 6) (9, 7, 5) (11, 9, 7, 5) (12, 10, 8) Compared to approximate solutions found by Concorde. As the number of points grows, the mean solution error first increases but then it settles to around 12.6%, while its standard deviation decreases. The solution error distributions are shown in Figure 8. Solution error distribution n=100 n=1000 n=10000 Solution error distribution n=200 n= Error [%] (a) Solution error distribution Error [%] n=500 n=5000 (b) Error [%] (c) Figure 8: Solution error distributions for random TSPs. The vertical lines mark the mean solution errors. 4.2 The 532 cities problem A popular TSPlib [43] benchmark for TSP solvers is a coordinate set of 532 non-uniformly distributed American cities (often abbreviated as att532). The points are depicted in Figure 9(a). The optimal solution was found by Padberg and Rinaldi [44] (scaled length of 10.08). The renormalization approach used by Yoshiyuki and Yoshiki [12] provided a solution which is 37% longer than the optimal one on average. A 10

11 simple memetic algorithm by Merz and Freisleben [45] yielded a solution approximately 20% longer than the optimal one, but a recombination-based version of this algorithm found the optimal solution. Ugajin s [13] moving-frame renormalization group approach yielded a solution that is 17% longer than the optimal one. A combination of local and global search operators by Tsai et al. found the optimal solution [46]. He et al. [47] used a parallel tabu search algorithm finding solutions 6% longer (on average) than the optimal one. Chen and Zhang [26] reported an enhanced annealing algorithm that found a solution 28% longer than the optimal one. Kalmár-Nagy and Giardini [33] used a genetic algorithm which provided very good solutions, with an average error of 1.1%. It is clear that most of the points of att532 are located in the top right quadrant of the unit square (348 out of 532 points). In order to keep the average size of clusters between 2-4 in this quadrant, the order of tilings in the tiling combination were chosen to be 9, 7 and 5. We solved this TSP instance times with randomly drawn tiling combinations. The mean solution error and the standard deviation of the error are listed in Table 2. Table 2: Mean and standard deviation of the solution error distribution for the att532 TSP. Number of points No. of TSPs Order of tilings Mean error [%] Std of error [%] (9, 7, 5) Figure 9(b) shows the solution error distribution compared to the results of n = 500 and 1000 i.i.d points. The solution error distribution for the att532 problem is similar to the n = 500 case, though the mean error is about 1.3% smaller. Solution error distribution n=500 att532 n= (a) Error [%] (b) Figure 9: Solution error distributions for the att532 problem and random TSPs. The vertical lines mark the mean solution errors. 4.3 Clustering with regular tilings A special type of dyadic tiling is the regular tiling which was used in the work of Yoshiyuki and Yoshiki [12]. A regular tiling consists of congruent squares, if the order of the tiling is even and congruent rectangles (whose sides are in the ratio 1:2) if the order of the tiling is odd. We solved TSPs using regular tilings for the n = 500 and 1000 cases. The mean solution error and the standard deviation of the error are listed in Table 3. 11

12 Table 3: Mean and standard deviation of the solution error distribution for random TSPs using regular tilings. Number of points No. of TSPs Order of tilings Mean error [%] Std of error [%] (7, 5, 3) (8, 6) Compared to approximate solutions found by Concorde. The solution error distributions together with those using random dyadic tilings are shown in Figure 10. Solution error distribution Solution error distribution Random dyadic tiling Regular tiling GA enhanced method Random dyadic tiling Regular tiling GA enhanced method Error [%] (a) Error [%] (b) Figure 10: The solution error distribution using random dyadic tilings and regular tilings for TSPs consisting of (a) n = 500 and (b) n = 1000 points. The vertical lines mark the mean solution errors. The difference between the mean solution error using random dyadic tilings and regular tilings is practically negligible: 0.56% and 0.44% for n = 500 and 1000 points, respectively. The result for the att532 problem is similar: the use of regular tilings yielded a solution 10.72% longer than the optimal. This is almost the same (0.2% longer) as the mean solution error for random dyadic tilings (10.52 %). Why then should one use dyadic tilings? Because the recombination of tilings can further improve solutions, as shown in the next section. To demonstrate the possible yield, in Figure 10 we also depict the results using the genetic algorithm enhanced hierarchical solution which is discussed in Section 5. 5 GA Enhanced Hierarchical Solution Even though the quality of solutions is quite satisfactory with the random tilings, a simple modification can yield even better results. We propose the Genetic Algorithm Enhanced Hierarchical Solution (GAEHS) method to improve the tiling combination. Genetic algorithms are a family of approximate optimization algorithms inspired by Darwin s theory of evolution [31, 32, 33]. In this context a possible tiling combination is called an individual (or plan). An individual is defined by a sequence of genes (these are the nodes of the HV -trees). In the initialization phase an initial plan is generated, similarly to the Genetic Algorithm Inspired Descent introduced by Kalmár-Nagy et al. [48] (in which there is only one initial team plan for the agents). This plan is evolved through a number of generation steps (the evolution phase). At every generation step the genes are recombined with genetic operators to create a new tiling combination. Then, the cost (the cost of a tiling combination is the length of the tour it determines) of the original and modified tiling combinations are evaluated. The flow chart diagram of GAEHS is shown in Figure

13 Initial solution Final solution No Yes The max no. of generations is reached? Keep the better solution Substitution operator (P substitution ) Mutation operator (P mutation ) Figure 11: Flow chart diagram of GAEHS. Evaluate the new solution The substitution operator (applied with probability P substitution ) substitutes a randomly chosen tiling of the current tiling combination with another random dyadic tiling. The mutation operator (applied with probability P mutation ) chooses one tiling of the current tiling combination and modifies it by randomly flipping (changing an H-node to V-node and vice versa) 3 nodes of the corresponding HV -tree. The better of the original and the modified tiling combination is kept. This process is repeated N times (number of generations). 5.1 Results with GAEHS We tested the GAEHS on random TSPs with n =100, 200, 500, and 1000 points, the number of generations was N = The probabilites were P substitution = 0.4 and P mutation = 0.8. The solution error after the initialization, the 100th and the 1000th generation was computed, the results are summarized in Table 4. Table 4: The mean error of the initial solutions and the solutions after 100 and 1000 generation steps. Number of points Number of TSPs Order of tilings Mean error of the initial solution [%] Mean error after 100 generations [%] Mean error after 1000 generations [%] (5, 3) (6, 4, 2) (7, 5, 3) att (9, 7, 5) (8, 6) Compared to approximate solutions found by Concorde. Figure 12 shows the solution error versus the number of generations for each n and the att532 problem (averaged result of the simulations). 13

14 Solution error [%] n=1000 n=500 att532 n=200 n= Number of generations N Figure 12: Evolution of the solution error through N = 1000 generation steps. The mean error curve shows how the solution error converged on average during the optimization procedure. The results already show significant improvement after 100 generations compared to the initial solutions or to those obtained with random tiling combinations. 6 Conclusions We solved the TSP with a new partition-and-cluster heuristic. We utilized random dyadic tilings to partition the TSP. Good quality subsolutions were computed in each tile, then they were patched together to yield the final solution. The method was tested for random TSPs with different n s (up to n = points), and for the att532 problem. The results support the viability of our method. The mean errors varied between % depending on n. For the att532 problem, our mean solution is considerably shorter (26.48%) than Yoshiyuki s solution (their approach was similar in spirit). We also tested the n = 500 and 1000 cases with only regular tilings used for partitioning. The solution error distributions (see Figure 10) are very similar to those obtained with random dyadic tilings. However, a random dyadic tiling can be modified to improve solution quality. We were able to significantly improve our initial results (e.g. the mean error for the n = 200 case was reduced to 3.97% from 10.37%) with an optimization algorithm called GAEHS, which looks for shorter tours by modifying the tiling combination. In future work incomplete HV -trees could be considered. This would vastly increase the possible number of tiling combinations. In addition, it would be advantageous to use incomplete HV -trees to partition nonuniformly distributed points (e.g. fine partitioning is required only in regions which are densely populated with points). GAEHS could be further improved by incorporating the Simulated Annealing technique. Another way to reduce the solution error is to identify the characteristics of dyadic tilings (e.g. by the distribution of the aspect ratio of tiles) and look for correlation among those and the solution error. If any correlation is found, more tilings which lead to better quality solutions could be generated (a similar approach for a different problem is discussed in [49]). Acknowledgements The authors thank Giovanni Giardini (SPS Italiana Pack Systems) for useful initial work on this project. This project was supported by the ÚNKP-17-3-I New National Excellence Program of the Ministry of Human Capacities of Hungary. We acknowledge the financial support from TeMA Talent Management Foundation. 14

15 References [1] C. H. Papadimitriou. The euclidean travelling salesman problem is NP-complete. Theoretical Computer Science, 4(3): , [2] G. Laporte. The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(2): , [3] D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook. The traveling salesman problem: a computational study. Princeton university press, [4] R. Matai, S. P. Singh, and M. L. Mittal. Traveling salesman problem: An overview of applications, formulations, and solution approaches. Traveling Salesman Problem, Theory and Applications, pages 1 24, [5] T. Volgenant and R. Jonker. A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation. European Journal of Operational Research, 9(1):83 89, [6] E. Balas and P. Toth. Branch and bound methods for the traveling salesman problem. Technical report, Carnegie-Mellon University, Pittsburgh, PA., [7] C. Rego, D. Gamboa, F. Glover, and C. Osterman. Traveling salesman problem heuristics: Leading methods, implementations and latest advances. European Journal of Operational Research, 211(3): , [8] J. J. Bartholdi and L. K. Platzman. An O (N log N) planar travelling salesman heuristic based on spacefilling curves. Operations Research Letters, 1(4): , [9] J. J. Bartholdi and L. K. Platzman. Heuristics based on spacefilling curves for combinatorial problems in Euclidean space. Management Science, 34(3): , [10] D. Bertsimas and M. Grigniy. On the space filling curve heuristic for the Euclidean traveling salesman problem. Technical report, Massachusetts Institute of Technology, Cambridge, MA., [11] R. M. Karp. Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Mathematics of operations research, 2(3): , [12] U. Yoshiyuki and K. Yoshiki. New method of solving the traveling salesman problem based on real space renormalization theory. Physical Review Letters, 75(9):1683, [13] R. Ugajin. Method to solve the travelling salesman problem using the inverse of diffusion process. Physica A: Statistical Mechanics and its Applications, 307(1): , [14] H. Braun. On solving travelling salesman problems by genetic algorithms. In International Conference on Parallel Problem Solving from Nature, pages Springer, [15] J. Houdayer and O. C. Martin. Renormalization for discrete optimization. Physical Review Letters, 83(5):1030, [16] Y. Deng, Y. Liu, and D. Zhou. An improved genetic algorithm with initial population strategy for symmetric TSP. Mathematical Problems in Engineering, 2015, [17] M. Yamada. 1/ f noise in the simple genetic algorithm applied to a traveling salesman problem. Fluctuation and Noise Letters, 16(03). [18] J. L. Bentley. Experiments on traveling salesman heuristics. In Proceedings of the first annual ACM- SIAM symposium on Discrete algorithms, pages Society for Industrial and Applied Mathematics, [19] M. Matayoshi, M. Nakamura, and H. Miyagi. A genetic algorithm with the improved 2-opt method. In Systems, Man and Cybernetics, 2004 IEEE International Conference on, volume 4, pages IEEE,

16 [20] S. Lin and B. W. Kernighan. An effective heuristic algorithm for the traveling-salesman problem. Operations research, 21(2): , [21] K. Helsgaun. An effective implementation of the lin kernighan traveling salesman heuristic. European Journal of Operational Research, 126(1): , [22] D. Applegate, W. Cook, and A. Rohe. Chained lin-kernighan for large traveling salesman problems. INFORMS Journal on Computing, 15(1):82 92, [23] M. Dorigo and T. Stützle. Ant colony optimization: overview and recent advances. In Handbook of metaheuristics, pages Springer, [24] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220(4598): , [25] T. J. P. Penna. Traveling salesman problem and Tsallis statistics. Physical Review E, 51(1):R1, [26] Y. Chen and P. Zhang. Optimized annealing of traveling salesman problem from the nth-nearest-neighbor distribution. Physica A: Statistical Mechanics and its Applications, 371(2): , [27] D. S. Johnson and L. A. McGeoch. The traveling salesman problem: A case study in local optimization. Local search in combinatorial optimization, 1: , [28] Z. Xiang, Z. Chen, X. Gao, X. Wang, F. Di, L. Li, G. Liu, and Y. Zhang. Solving large-scale TSP using a fast wedging insertion partitioning approach. Mathematical Problems in Engineering, 2015, [29] J. C. Lagarias, J. H. Spencer, and J. P. Vinson. Counting dyadic equipartitions of the unit square. Discrete Mathematics, 257(2-3): , [30] S. Janson, D. Randall, and J. Spencer. Random dyadic tilings of the unit square. Random Structures & Algorithms, 21(3-4): , [31] D. E. Goldberg and J. H. Holland. Genetic algorithms and machine learning. Machine learning, 3(2):95 99, [32] K. Katayama, H. Sakamoto, and H. Narihisa. The efficiency of hybrid mutation genetic algorithm for the travelling salesman problem. Mathematical and Computer Modelling, 31(10): , [33] G. Giardini and T. Kalmár-Nagy. Genetic algorithm for combinatorial path planning: the subtour problem. Mathematical Problems in Engineering, 2011, Special section:1 31, [34] G. G. Lai, D. Fussell, and D. F. Wong. HV/VH trees: a new spatial data structure for fast region queries. In Design Automation, th Conference on, pages IEEE, [35] C. J Alpert, D. P. Mehta, and S. S. Sapatnekar. Handbook of algorithms for physical design automation. CRC press, [36] A. Skodras, C. Christopoulos, and T. Ebrahimi. The JPEG 2000 still image compression standard. IEEE Signal processing magazine, 18(5):36 58, [37] S. Weyer, A. Fröhlich, H. Riesch-Oppermann, L. Cizelj, and M. Kovac. Automatic finite element meshing of planar Voronoi tessellations. Engineering Fracture Mechanics, 69(8): , [38] J. Beardwood, J. H. Halton, and J. M. Hammersley. The shortest path through many points. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 55, pages Cambridge Univ Press, [39] J. L. Jacobsen, N. Read, and H. Saleur. Traveling salesman problem, conformal invariance, and dense polymers. Physical review letters, 93(3):038701,

17 [40] D. S. Johnson, L. A. McGeoch, and E. E. Rothberg. Asymptotic experimental analysis for the Held Karp traveling salesman bound. In Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms, volume 81, page 341. SIAM, [41] A. G. Percus and O. C. Martin. Finite size and dimensional dependence in the euclidean traveling salesman problem. Physical Review Letters, 76(8):1188, [42] S. Steinerberger. New bounds for the traveling salesman constant. Advances in Applied Probability, 47(01):27 36, [43] G. Reinelt. TSPLIB - A traveling salesman problem library. ORSA journal on computing, 3(4): , [44] M. Padberg and G. Rinaldi. Optimization of a 532-city symmetric traveling salesman problem by branch and cut. Operations research letters, 6(1):1 7, [45] P. Merz and B. Freisleben. Memetic Algorithms for the Traveling Salesman Problem. Complex Systems, 13(4): , [46] H. K. Tsai, J. M. Yang, and C. Y. Kao. Solving traveling salesman problems by combining global and local search mechanisms. Evolutionary Computation, CEC 02. Proceedings of the 2002 Congress on, 2, [47] H. Yi, L. Guangyuan, and Q. Yuhui. A parallel tabu search algorithm based on partitioning principle for TSPs. International Journal of Computer Science and Network Security, 6(8A):146, [48] T. Kalmár-Nagy, G. Giardini, and B. D. Bak. The multi-agent planning problem. Complexity, 2017:Article ID , 12 pages, [49] T. Shaikhina and N. A. Khovanova. Handling limited datasets with neural networks in medical applications: A small-data approach. Artificial Intelligence in Medicine, 75:51 63,