Hierarchical Solution of the Traveling Salesman Problem with
Random Dyadic Tilings
Tamas Kalmar-Nagy∗ and Bendeguz Dezso Bak†
Department of Fluid Mechanics, Faculty of Mechanical EngineeringBudapest University of Technology and Economics
Abstract
We propose a hierarchical heuristic approach for solving the Traveling Salesman Problem (TSP) inthe unit square. The points are partitioned with a random dyadic tiling and clusters are formed by thepoints 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 inthe coarse solution. Near-optimal (or optimal) minimum tours are computed for each cluster. The toursare concatenated using the midpoints yielding a solution for the original TSP. The method is tested onrandom TSPs independent, identically distributed points in the unit square) up to 10000 points as well ason 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 improveour solutions. This algorithm further reduces the solution errors (by several percent using 1000 iterationsteps). 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 tostate, yet hard to solve combinatorial problem: given n points, find the minimum length tour connecting allpoints that starts and ends at the same point. The TSP is an NP-hard problem [1]. Thorough studies ofthe TSP are given by Laporte [2] and Applegate et. al [3], the latter includes vast amount of computationalresults. Another overview of exact and approximate methods is given by Matai et al. [4], as well as examplesof 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 thebranch-and-cut method [3]. The main drawback of these (and other) exact methods is the extensive amountof 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 eithertour construction or tour improvement type methods. Tour construction methods gradually build up theTSP solution. Approximate methods which rely on the use of spacefilling curves [8, 9, 10] and partitioningalgorithms [11, 12, 13] are tour construction methods. Tour improvement methods generate an initial tourand try to alter this tour to become shorter. Tour improvement using genetic algorithm [14, 15, 16, 17] is apopular approximate method. Another tour improvement type method is the 2-opt method [18, 19] whichreorders the TSP solution to reduce tour length. It takes a section from the tour and adds it in reverse orderto form a new tour as shown in Figure 1.
∗[email protected]†[email protected]
1
1 2
3
4
6
57
(a)
1 2
3
4
6
57
(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 becomesshorter (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 currentlyconsidered one of the best approximate methods for the TSP. The major drawback of approximate methodsis that they often get stuck at a local minimum. This can be prevented using a metaheuristic such as antcolony optimization [23] or simulated annealing [24, 25, 26]. An overview of these metaheuristics is given byJohnson et al. [27].
Focusing on partitioning algorithms, an influential piece of work by Karp [11] considers such algorithms forthe approximate solution of large instances of the TSP. To reduce the complexity of the problem, partitioningalgorithms subdivide the set of points into small groups, construct an optimum tour through each group, andthen 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 eachcluster, (d) and these are concatenated.
First the starting set of points (depicted in Figure 2(a)) is “reduced”: the points are grouped into disjointsubsets (clusters) as shown in Figure 2(b). Good quality subsolutions are computed for each cluster ofpoints (Figure 2(c)). A cluster of points is replaced with a representative point, and the subsolutions areconcatenated (Figure 2(d)).
Yoshiyuki and Yoshiki [12] introduced a method in which the unit square was recursively subdividedinto smaller squares (regular square tiling). After each division the points located in the same subsquarewere substituted with the barycenter of that subsquare. For every subdivision an approximate solutionwas 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 tilesare 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 agenetic algorithm.
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In this work, we propose a new partition-and-cluster approach. The set of points is partitioned withdyadic tilings [29, 30], yielding clusters of points. A TSP solution is calculated for the barycenters of theclusters. A tour with different endpoints (subsolution) is calculated for the points of each cluster. Thesetours are concatenated based on the TSP solution of the barycenters, yielding the global solution. The globalsolution strongly depends on the tiling, but this dependence is unknown a priori. A large number of TSPsconsisting of n independent, identically distributed points in the unit square is solved using random dyadictilings to investigate the solution quality.
To improve the efficacy of the algorithm we introduce the Genetic Algorithm Enhanced HierarchicalSolution (GAEHS) method which borrows elements of the classical genetic algorithm [31, 32, 33]. Thegenetic operators randomly choose and modify parts of the tiling. The modified tiling is kept if it is betterthan its predecessor.
This paper is structured as follows: in Section 2 we describe dyadic tiling, its connection with labeledcomplete binary trees and how the set of points is partitioned. In Section 3 we explain how a global solutionof the TSP can be constructed in a hierarchical manner. In Section 4 results are presented for random TSPsand 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 partitioningof the plane to find objects intersecting a specified area is a common approach [34, 35]. Tiling has beenapplied 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 lthorder dyadic tiling partitions the unit square into 2l tiles of area 2−l [29, 30]. There is a correspondencebetween dyadic tilings and labeled complete binary trees (HV -trees) [30, 34]. Figure 3 illustrates such acorrespondence.
(a) (b) (c)
Figure 3: (a) A 2nd order dyadic tiling, (b) illustration of the subdivision process, and (c) the correspondingHV -tree.
Figure 3(b) illustrates the “cutting” process. The unit square is depicted on the top. The labels H and Vindicate whether the tile above them was halved horizontally or vertically. The loose edges at the bottomcorrespond to the tiles shown next to them. These tiles constitute the tiling shown in Figure 3(a). If the
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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 thedyadic tiling. For a given HV -tree the 4-tuple (a, b, s, t) can be calculated for each tile. If the 4-tuple of anlth 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),
(3)
where HL, HR, VL, VR are R4 7→ R4 affine transformations. Although every HV -tree uniquely defines thecorresponding dyadic tiling, the reverse is not true. The number of HV -trees which represent every distinctlth order dyadic tiling is given by the recurrence [30]
Nl = 2N2l−1 −N4
l−2, N0 = 1, N1 = 2. (4)
The first few elements of the sequence are
N2 = 7, N3 = 82, N4 = 11047, N5 = 198860242, . . . . (5)
2.2 Tessellation and partitioning
Let HV(l) denote the set of lth order HV -trees. A dyadic tiling T ∈ HV(l) provides a natural tessellationof the unit square. Given a set of points X in the unit square, let T (X) denote the set of clusters obtainedby grouping points belonging to the same tile. This is a set theoretical partitioning, i.e. the elements of Xare grouped into non-empty subsets such that every element is included in exactly one of the subsets. Thesize of a cluster is the number of points it contains. Figure 4 shows a dyadic tiling and the correspondingclusters (encircled).
Figure 4: The unit square is partitioned with a dyadic tiling. The clusters of points (dots) are encircled. Thebarycenters (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 uniquelydetermines its tile. Barycenters are clustered similarly to the original points.
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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 followingsteps 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 ofbarycenters 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 placedat the middle of each edge. Two midpoints naturally belong to each barycenter (located on the edgesemanating from the barycenter).
3. Each barycenter is replaced by the cluster of points it represents (these points are located in its tileas 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. Theunion of all such tours forms a closed tour. Figure 5(c) highlights each tour with alternating continuousand dashed lines.
4. Midpoints × are removed from the tour to yield the TSP solution for the original points. At this pointthe 2-opt method is used to improve the final solution (Figure 5(d)).
5
(a) (b)
(c) (d)
Figure 5: (a) Partitioning the points, (b) “coarse” TSP solution for the barycenters and placement of themidpoints, (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 onemust deal with the following issues:
• If only a few tiles are used for partitioning, there are too many points in each cluster. This makes tourconstructions in Step 3 computationally slow.
• If the order of the tiling is high, there will be too many barycenters. In this case the construction ofthe 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 clustermust also be small, i.e. both the number of clusters and the average number of points in each cluster mustbe 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 T1, so that the average cluster size is sufficiently small (preferably less than
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10 points). This yields the set of clusters C1 = T1(X) and the set of barycenters B1. As discussed, in thiscase the number of barycenters will be too high (usually their number is in the same order of magnitude asn) to efficiently construct the “coarse” TSP solution. Instead, the barycenters are partitioned with anothertiling T2 which is coarser than T1 (i.e. we obtain C2 = T2(B1) and B2). The new barycenters are furtherpartitioned with gradually coarser tilings T3, ..., Ti, Ti+1, ..., Tk. The number of barycenters in Bk is assumedto 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 Bi−1 is built from that of Bi. The TSP solution for theoriginal points is built from that of B1. The set of tilings (T1, T2, ...., Tk) used for the partitioning is calledthe tiling combination.
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𝒯1 𝑋
𝒯2 𝐵1
keep add ×
keep ×
keep +
Remove +use 2-opt
Remove ×2-optadd +
Figure 6: Hierarchical solution of the TSP using two partitionings. Legend: • are the points, � are thebarycenters of C1 and 4 are the barycenters of C2, The midpoints are denoted by × and +.
In Figure 6 a solution is shown step-by-step using two tilings: T1 and T2. X is partitioned with T1 whichyields the set of barycenters B1 (squares). B1 is partitioned further with T2 (which is coarser than T1) yieldingthe set of barycenters B2 (triangles). The “coarse” TSP solution is computed for B2 and midpoints × areplaced. To every barycenter 4 belong two midpoints × and its barycenters � from the first partitioning.The TSP solution for B1 is built from that of B2. The midpoints × are removed during the process. Newmidpoints + are added to the middle of each edge in the TSP solution for B1. Two midpoints + and acluster of points belong to each barycenter �. The TSP solution for the original points is built from that ofB1. The midpoints + are removed during the process.
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4 Computational Results
The hierarchical algorithm of Section 3 was tested for n independent, identically distributed points in theunit square. The tiling combinations were chosen so as to keep the average size of clusters around 2-4, i.e.2 ≤ n/2l ≤ 4, where l is the order of the first tiling. For instance, 500 points are first partitioned with a 7thorder tiling, thus the average number of points in each cluster is 500/27 ≈ 3.91. The orders of the subsequenttilings in the tiling combination are l − 2, l − 4... (the orders of the subsequent tilings are 5 and 3 for 500points). 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 Lmin for n independent,identically distributed points in the unit square behaves as
limn→∞
Lmin/√n = β, (6)
where β ≈ 0.7119 [39] is the Beardwood-Halton-Hammersley constant. Many studies dealt with the numericalapproximation of β [39, 40, 41] or the determination of its lower and upper bounds [42]. The approximation
Lmin = β√n+ C. (7)
is consistent with Eq. (6). Using Table 16.7 from Chapter 16 of [3] yielded β = 0.71 and C = 0.63. Figure 7shows Lmin (solid line), and the mean lengths of our solutions (circles) for various n′s.
Mean solution length
0.71 n +0.63
2000 4000 6000 8000 10000
20
40
60
80
Solution length
Number of points n
Figure 7: The Lmin = 0.71√n+0.63 curve and the mean tour lengths obtained with our hierarchical method.
TSP solutions were computed for n =100-10000 independent, identically distributed points in the unit square.The “coarse” TSP solution was always calculated exactly (by brute force or Concorde). Exact solutions werecalculated 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 TSPsolutions were calculated using Concorde. The solution error shows how much longer is (on the average) thetour we found compared to the optimal/near-optimal one.
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Table 1: Mean and standard deviation of the solution error distribution for random TSPs.
Numberof points
No. ofTSPs
Orderof tilings
Meanerror [%]
Std oferror [%]
100 10000 (5, 3) 8.89 2.85200 10000 (6, 4, 2) 10.50 2.16500 10000 (7, 5, 3) 11.81 1.501000 1000 (8, 6) 12.361 1.082000 1000 (9, 7, 5) 12.601 0.805000 100 (11, 9, 7, 5) 12.651 0.5410000 100 (12, 10, 8) 12.651 0.38
1 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.
n=100
n=1000
n=10000
0 5 10 15 20
0.2
0.4
0.6
0.8
1.0
Solution error
distribution
Error [%]
(a)
n=200
n=2000
4 6 8 10 12 14 16 18
0.1
0.2
0.3
0.4
0.5
0.6
Solution error
distribution
Error [%]
(b)
n=500
n=5000
6 8 10 12 14 16 18
0.2
0.4
0.6
0.8
Solution error
distribution
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 distributedAmerican cities (often abbreviated as att532). The points are depicted in Figure 9(a). The optimal solutionwas found by Padberg and Rinaldi [44] (scaled length of 10.08). The renormalization approach used byYoshiyuki and Yoshiki [12] provided a solution which is 37% longer than the optimal one on average. A
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simple memetic algorithm by Merz and Freisleben [45] yielded a solution approximately 20% longer than theoptimal 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. Acombination 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. Chenand Zhang [26] reported an enhanced annealing algorithm that found a solution 28% longer than the optimalone. Kalmar-Nagy and Giardini [33] used a genetic algorithm which provided very good solutions, with anaverage 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 (348out of 532 points). In order to keep the average size of clusters between 2-4 in this quadrant, the order oftilings in the tiling combination were chosen to be 9, 7 and 5. We solved this TSP instance 10000 times withrandomly drawn tiling combinations. The mean solution error and the standard deviation of the error arelisted in Table 2.
Table 2: Mean and standard deviation of the solution error distribution for the att532 TSP.
Numberof points
No. ofTSPs
Orderof tilings
Meanerror [%]
Std oferror [%]
532 10000 (9, 7, 5) 10.52 1.41
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 erroris about 1.3% smaller.
(a)
n=500
att532
n=1000
6 8 10 12 14 16 18
0.1
0.2
0.3
0.4
Solution error
distribution
Error [%]
(b)
Figure 9: Solution error distributions for the att532 problem and random TSPs. The vertical lines mark themean 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 forthe n = 500 and 1000 cases. The mean solution error and the standard deviation of the error are listed inTable 3.
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Table 3: Mean and standard deviation of the solution error distribution for random TSPs using regulartilings.
Numberof points
No. ofTSPs
Orderof tilings
Meanerror [%]
Std oferror [%]
500 10000 (7, 5, 3) 12.37 2.081000 1000 (8, 6) 12.891 1.09
1 Compared to approximate solutions found by Concorde.
The solution error distributions together with those using random dyadic tilings are shown in Figure 10.
Random dyadic tiling
Regular tiling
GA enhanced method
6 8 10 12 14 16 18 20
0.1
0.2
0.3
0.4
0.5
0.6
Solution error
distribution
Error [%]
(a)
Random dyadic tiling
Regular tiling
GA enhanced method
8 10 12 14 16 18
0.2
0.4
0.6
0.8
Solution error
distribution
Error [%]
(b)
Figure 10: The solution error distribution using random dyadic tilings and regular tilings for TSPs consistingof (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 practicallynegligible: 0.56% and 0.44% for n = 500 and 1000 points, respectively. The result for the att532 problem issimilar: 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 improvesolutions, as shown in the next section. To demonstrate the possible yield, in Figure 10 we also depict theresults 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 modificationcan 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 ofevolution [31, 32, 33]. In this context a possible tiling combination is called an individual (or plan). Anindividual is defined by a sequence of genes (these are the nodes of the HV -trees). In the initialization phasean initial plan is generated, similarly to the Genetic Algorithm Inspired Descent introduced by Kalmar-Nagyet al. [48] (in which there is only one initial team plan for the agents). This plan is evolved through a numberof generation steps (the evolution phase). At every generation step the genes are recombined with geneticoperators to create a new tiling combination. Then, the cost (the cost of a tiling combination is the lengthof the tour it determines) of the original and modified tiling combinations are evaluated. The flow chartdiagram of GAEHS is shown in Figure 11.
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Initial solution
Substitution operator (𝑃𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛)
Evaluate thenew solution
Mutation operator (𝑃𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛)
Keep the bettersolution
The max no. of generations is reached?
No
Yes
Final solution
Figure 11: Flow chart diagram of GAEHS.
The substitution operator (applied with probability Psubstitution) substitutes a randomly chosen tiling of thecurrent tiling combination with another random dyadic tiling.
The mutation operator (applied with probability Pmutation) chooses one tiling of the current tiling com-bination and modifies it by randomly flipping (changing an H-node to V-node and vice versa) 3 nodes of thecorresponding HV -tree. The better of the original and the modified tiling combination is kept. This processis 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 generationswas N = 1000. The probabilites were Psubstitution = 0.4 and Pmutation = 0.8. The solution error after theinitialization, 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.
Numberof points
Numberof TSPs
Orderof tilings
Mean errorof the initialsolution [%]
Mean errorafter 100
generations [%]
Mean errorafter 1000
generations [%]
100 100 (5, 3) 8.51 2.71 1.48200 100 (6, 4, 2) 10.37 5.61 3.97500 100 (7, 5, 3) 11.71 8.28 7.06
att532 20 (9, 7, 5) 10.33 7.02 5.801000 20 (8, 6) 12.301 9.721 8.761
1 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).
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1 5 10 50 100 500 1000
2
4
6
8
10
12
Solution error [%]
n=1000
n=500
att532
n=200
n=100
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 orto 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 partitionthe TSP. Good quality subsolutions were computed in each tile, then they were patched together to yieldthe final solution. The method was tested for random TSPs with different n’s (up to n = 10000 points), andfor the att532 problem. The results support the viability of our method. The mean errors varied between8.89-12.65% 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 caseswith only regular tilings used for partitioning. The solution error distributions (see Figure 10) are verysimilar to those obtained with random dyadic tilings. However, a random dyadic tiling can be modified toimprove solution quality. We were able to significantly improve our initial results (e.g. the mean error forthe n = 200 case was reduced to 3.97% from 10.37%) with an optimization algorithm called GAEHS, whichlooks for shorter tours by modifying the tiling combination.
In future work incomplete HV -trees could be considered. This would vastly increase the possible numberof tiling combinations. In addition, it would be advantageous to use incomplete HV -trees to partition non-uniformly distributed points (e.g. fine partitioning is required only in regions which are densely populatedwith 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 thedistribution of the aspect ratio of tiles) and look for correlation among those and the solution error. If anycorrelation is found, more tilings which lead to better quality solutions could be generated (a similar approachfor 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 UNKP-17-3-I New National Excellence Program of the Ministry of HumanCapacities of Hungary. We acknowledge the financial support from TeMA Talent Management Foundation.
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