Helsgaun, K.: Solving the Equality Generalized Traveling Salesman Problem Using the Lin-Kernighan-Helsgaun Algorithm. Computer Science Report #141, Ro...

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Abstract It is well known that many arc routing problems can be transformed into the Equality Generalized Traveling Salesman Problem (E-GTSP), which in turn can be transformed into a standard Asymmetric Traveling Salesman Problem (TSP). This opens up the possibility of solving arc routing problems using existing solvers for TSP. This paper evaluates the performance of the state-of-the art TSP solver Lin-Kernighan-Helsgaun (LKH) on a broad class of transformed arc routing instances. It is shown that LKH makes it possible to find solutions of good quality to large-scale undirected, mixed, and windy postman and general routing problem instances. Keywords: Arc routing problems, Equality generalized traveling salesman problem, E-GTSP, Traveling salesman problem, TSP Mathematics Subject Classification: 90C27, 90C35, 90C59

1. Introduction The goal of arc routing problems (ARPs) is to determine a minimum cost closed walk passing through some arcs and edges of a graph. Formally, ARPs are defined on a graph G = (V, A, E) where V = {v1, ..., vn} is a set of vertices, A is a set of (directed) arcs aij (i ≠ j), and E is a set of (undirected) edges eij (i < j). Non-negative costs cij and dij are associated with arcs aij and with edges eij, respectively. It is not necessary to traverse all arcs or edges. Denote by AR and ER the subsets of required arcs and edges, respectively. The aim is to determine a least cost closed walk on G including all required arcs and edges at least once. If the walk also has to pass through a certain subset of required vertices, VR ⊆ V, we have the general routing problem (GRP). Depending on problem properties, some well-known classes of routing problems can be obtained from this definition. In this paper the following classes will be tackled: • The Mixed Chinese Postman Problem (MCPP): AR = A ≠ ∅, ER = E ≠ ∅, dij = dji for all i, j, VR =∅. • The Windy Postman Problem (WPP): A =∅, ER = E ≠ ∅, dij ≠ dji for at least one edge eij, VR =∅.

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• The Undirected, Mixed and Windy Rural Postman Problems (URPP, MRPP, WRPP), which are defined similarly, except that now AR ⊂ A or ER ⊂ E. • The General Routing Problem (GRP): A = ∅, ER = E ≠ ∅, dij = dji for all i, j, VR ≠ ∅. • The Mixed General Routing Problem (MGRP): A =∅, ER = E ≠ ∅, dij = dji for all i, j, V R ≠ ∅. • The Windy General Routing Problem (WGRP): A =∅, ER = E ≠ ∅, dij ≠ dji for at least one edge eij, VR ≠ ∅. All these problems can easily be transformed into E-GTSP [1]. The transformed problem is defined on a graph H = (W, B). In this graph W consists of one vertex wij for each required arc vij of G, one vertex wii for each required vertex vi, and two vertices wij and wji for each required edge eij (one for each of the corresponding opposite arcs, only one of which is required). B is the set of all arcs linking two vertices of W. Each vertex pair (wki, wlj) in the transformed problem defines an arc of W with a cost equal to sil + clj, where sil denotes the cost of a shortest path from vi to vl on G. We have thus transformed the original arc routing problem into the Equality Generalized Traveling Salesman Problem (E-GTSP), where each cluster consists of either one or two vertices. Clusters consisting of two vertices correspond to required edges in the original problem, whereas single vertex clusters correspond to required arcs and required vertices in the original problem. A recent paper [2] has described GLKH, an effective solver for E-GTSP based on the LinKernighan-Helsgaun algorithm, LKH [3]. GLKH will be used in the following computational study. 2. Computational Results The program was coded in C and run under Linux on an iMac 3.4 GHz Intel Core i7 with 32 GB RAM. Version 1.0 of GLKH was used. The program uses only one of the computer’s four CPU cores. Coberán et al. have provided a large library of test instances for arc routing problems [4]. The library includes 1042 instances of URPP, GRP, MCPP, MRPP, MGRP, WPP, WRPP and WGRP. All these instances have be transformed into E-GTSP and then solved by GLKH using the following non-default parameter settings: ASCENT_CANDIDATES = 500 INITIAL_PERIOD = 1000 MAX_CANDIDATES = 12 MAX_TRIALS = 1000 OPTIMUM =

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The following observations can be made: •

The solution quality is good for all instances. Optima are found for about half of the instances and the average deviation from the optimal solution is less than 3%.

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The instances in MCPP and WPP are the most difficult for GLKH. Other parameter settings might lead to a better solution quality. However, this will probably be at the expense of unacceptable running times. For these large instances, GLKH cannot compete with the highly sophisticated exact algorithm of Corberán et al. [5].

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Tests similar to those reported in Table 1 have been conducted by Drexl [6, p. 10]. Using Gutin and Karapetyan’s heuristic E-GTSP solver GK [7], he found that GK performed acceptable for instances with up to about 200 clusters. However, for instances with more than 500 clusters, the gap to the optimal solutions usually exceeded 10%. As seen, GLKH performs better than GK for instances with many clusters.

Currently, optima are known for 998 out of the 1042 instances. It may be mentioned, that until now GLKH has been able to find new best upper bounds for 22 of the remaining 44 instances: MCPP MA3067 6,529,588 MCCP MB2052: 125,566 MCPP MB3052: 151,284 MCPP MB3065: 201,187 MGRP GD422: 32,057 MGRP GD425: 37,581 MGRP GD522: 34,482 MGRP GD525: 40,077 WPP WA3065: 4,500,431 WPP WB3035: 83,596 WPP WB3055: 133,501

WPP WB3061: 178,684 WPP WB3062: 177,765 WRPP C422: 21,181 WRPP D322: 23,784 WRPP D421: 24,539 WRPP D422: 23,943 WGRP GB321: 20,549 WGRP GB322: 20,328 WGRP GB421: 20,774 WGRP GB422: 20,452 WGRP GB622: 24,102

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Instance classes URPP: UR500 URPP: UR750 URPP: UR1000 GRP: Alba GRP: Madr GRP: GRP MCPP: MA05 MCPP: MB05 MCPP: MA10 MCPP: MB10 MCPP: MA15 MCPP: MB15 MCPP: MA20 MCPP: MB20 MCPP: MA30 MCPP: MB30 MRPP: RB MRPP: RD MGRP: Alba MGRP: Alda MGRP: Madr MGRP: GB MGRP: GD WPP: WA05 WPP: WB05 WPP: WA10 WPP: WB10 WPP: WA15 WPP: WB15 WPP: WA20 WPP: WB20 WPP: WA30 WPP: WB30 WRPP: A100 WRPP: A500 WRPP: A1000 WRPP: B WRPP: C WRPP: D WRPP: M WRPP: HD WRPP: HG WRPP: P WGRP: A WGRP: G

# of inst. 12 12 12 15 15 10 12 12 12 12 12 12 12 12 12 12 18 18 25 31 25 18 18 12 12 12 12 12 12 12 12 12 12 72 27 27 24 24 24 72 54 54 144 27 24

|V| |A|+|E| 446 1129 666 1698 886 2290 116 174 196 316 116 174 500 1158 500 1210 1000 2319 1000 2442 1500 3479 1500 3631 2000 4645 2000 4829 3000 6959 3000 7131 449 1134 900 2315 116 174 214 351 196 316 500 1218 1000 2450 500 1160 500 1213 1000 2317 1000 2434 1500 3493 1500 3655 2000 4645 2000 4826 3000 6961 3000 7141 116 174 401 1268 848 2522 446 1132 673 1706 895 2287 196 316 86 173 83 149 25 59 500 1135 500 1210

|AR|+|ER| 616 907 1215 86 158 75 1158 1210 2319 2442 3479 3631 4645 4829 6959 7131 610 1230 88 168 158 610 1230 1160 1213 2317 2434 3493 3655 4645 4826 6961 7141 102 481 1149 610 918 1222 187 85 77 28 575 599

n 1232 1813 2430 196 347 178 1773 1836 3555 3702 5330 5511 7108 7329 10664 10877 1376 2759 177 324 301 980 1958 2321 2426 4634 4868 6986 7309 9289 9652 13922 14282 204 963 2297 1220 1837 2443 374 170 154 56 1223 1255

m Error (%) 616 0.32 907 0.40 1215 0.50 110 0.00 189 0.00 102 0.00 1158 0.56 1210 0.24 2319 0.86 2442 0.61 3479 1.06 3631 0.57 4645 1.09 4829 0.58 6959 1.20 7131 0.75 610 0.02 1230 0.08 118 0.00 217 0.00 205 0.00 661 0.03 1330 0.08 1160 2.00 1213 1.28 2317 2.44 2434 2.15 3493 2.68 3655 2.17 4645 2.86 4826 2.36 6961 2.97 7141 2.35 102 0.01 481 1.13 1149 1.71 610 0.28 918 0.40 1222 0.58 187 0.04 85 0.01 77 0.00 28 0.00 648 1.18 656 0.30

Table 1 Results for arc routing problems.

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Opt. (%) 41.7 25.0 25.0 100.0 100.0 100.0 8.3 25.0 0.0 16.7 0.0 8.3 0.0 0.0 0.0 0.0 72.2 33.3 100.0 93.5 100.0 83.3 44.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 94.4 3.7 0.0 12.5 8.3 12.5 68.1 96.3 100.0 100.0 0.0 12.5

Time (s) 141.1 231.4 358.4 0.2 1.5 0.2 1028.9 740.2 2958.7 2283.6 4686.8 3726.4 7031.0 5226.9 12627.1 8511.3 141.0 530.9 0.2 7.1 1.7 110.5 558.3 489.8 386.3 1230.2 899.4 2229.5 1678.3 3412.7 2561.3 7375.6 5060.1 1.1 134.8 505.4 125.2 170.8 251.1 10.8 0.9 0.6 0.0 191.1 124.8

3. Conclusion The computational results show that LKH makes it possible to find solutions of good quality to large-scale undirected, mixed, and windy postman and general routing problem instances. The developed software is free of charge for academic and non-commercial use and can be downloaded in source code together with test instances via http://www.ruc.dk/~keld/research/GLKH/.

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References 1. Blais, M., Laporte, G.: Exact Solution of the Generalized Routing Problem through Graph Transformations. J. Oper. Res. Soc., 54(8):906-910 (2003) 2. Helsgaun, K.: Solving the Equality Generalized Traveling Salesman Problem Using the Lin-Kernighan-Helsgaun Algorithm. Computer Science Report #141, Roskilde University (2014) 3. Helsgaun, K.: An Effective Implementation of the Lin-Kernighan Traveling Salesman Heuristic. Eur. J. Oper. Res., 126(1):106-130 (2000) 4. Corberán, Á., Plana I., Sanchis, J.M.: Arc Routing Problems: Data Instances. http://www.uv.es/corberan/instancias.htm 5. Corberán, A., Oswald, M., Plana I., Reinelt, G., Sanchis, J.M.: New results on the Windy Postman Problem, Math. Program., Ser. A 132:309–332 (2012) 6. Drexl, M.: On the generalized directed rural postman problem. J. Oper. Res. Soc., doi:10.1057/jors.2013.60 (2013) 7. Gutin, G., Karapetyan, D.: A memetic algorithm for the generalized traveling salesman problem. Nat. Comput., 9(1):47-60 (2010)

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