New Heuristic Algorithms for the Windy Rural Postman Problem

[EN] In this paper we deal with the windy rural postman problem. This problem generalizes several important arc routing problems and has interesting real-life applications. Here, we present several heuristics whose study has lead to the design of a scatter search algorithm for the windy rural postman problem. Extensive computational experiments over different sets of instances, with sizes up to 988 nodes and 3952 edges, are also presented. (c) 2004 Elsevier Ltd. All rights reserved.Benavent, E.; Corberán, A.; Piñana, E.; Plana. I.; Sanchís Llopis, JM. (2005). New Heuristic Algorithms for the Windy Rural Postman Problem. Computers & Operations Research. 32(12):3111-3128. doi:10.1016/j.cor.2004.04.007S31113128321

New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem

[EN] The min-max windy rural postman problem is a multiple vehicle version of the windy rural postman problem, WRPP, which consists of minimizing the length of the longest route to find a set of balanced routes for the vehicles. In a previous paper, an ILP formulation and a partial polyhedral study were presented, and a preliminary branch-and-cut algorithm that produced some promising computational results was implemented. In this article, we present further results for this problem. We describe several new facet-inducing inequalities obtained from the WRPP, as well as some inequalities that have to be satisfied by any optimal solution. We present an enhanced branch-and-cut algorithm that takes advantage of both these new inequalities and high quality min-max K-WRPP feasible solutions obtained by a metaheuristic. Computational results on a large set of instances are also reported. © 2011 Wiley Periodicals, Inc.Contract grant sponsor: Ministerio de Ciencia e Innovacion of Spain; Contract grant numbers: MTM2006-14961-C05-02, MTM2009-14039-C06-02Benavent López, E.; Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2011). New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem. Networks. 58(4):255-272. https://doi.org/10.1002/net.20469S255272584D. Ahr Contributions to multiple postmen problems 2004Ahr, D., & Reinelt, G. (2002). New Heuristics and Lower Bounds for the Min-Max k-Chinese Postman Problem. Lecture Notes in Computer Science, 64-74. doi:10.1007/3-540-45749-6_10Ahr, D., & Reinelt, G. (2006). A tabu search algorithm for the min–max k-Chinese postman problem. Computers & Operations Research, 33(12), 3403-3422. doi:10.1016/j.cor.2005.02.011D. Applegate R. E. Bixby V. Chvátal W. Cook Finding cuts in the TSP 1995Benavent, E., Carrotta, A., Corberán, A., Sanchis, J. M., & Vigo, D. (2007). Lower bounds and heuristics for the Windy Rural Postman Problem. European Journal of Operational Research, 176(2), 855-869. doi:10.1016/j.ejor.2005.09.021Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2009). Min-Max K -vehicles windy rural postman problem. Networks, 54(4), 216-226. doi:10.1002/net.20334Benavent, E., Corberán, Á., & Sanchis, J. M. (2009). A metaheuristic for the min–max windy rural postman problem with K vehicles. Computational Management Science, 7(3), 269-287. doi:10.1007/s10287-009-0119-2Corberáan, A., Letchford, A. N., & Sanchis, J. M. (2001). A cutting plane algorithm for the General Routing Problem. Mathematical Programming, 90(2), 291-316. doi:10.1007/pl00011426Corberán, A., Plana, I., & Sanchis, J. M. (2007). A branch & cut algorithm for the windy general routing problem and special cases. Networks, 49(4), 245-257. doi:10.1002/net.20176Corberán, A., Plana, I., & Sanchis, J. M. (2008). The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics, 22(2), 606-628. doi:10.1137/050640886Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017Pearn, W. L. (1994). Solvable cases of the k-person Chinese postman problem. Operations Research Letters, 16(4), 241-244. doi:10.1016/0167-6377(94)90073-6I. Plana The windy general routing problem 200

New results on the Windy Postman Problem

[EN] In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal-Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP. © 2010 Springer and Mathematical Optimization Society.The authors want to thank the three referees for their careful reading of the manuscript and for their many comments and suggestions that have contributed to improve the paper content and readability. In particular, several remarks regarding the discussion of C-G and mod-k inequalities were pointed out by one of the referees. A. Corberan, I. Plana and J.M. Sanchis wish to thank the Ministerio de Educacion y Ciencia of Spain (projects MTM2006-14961-C05-02 and MTM2009-14039-C06-02) for its support.Corberán, A.; Oswald, M.; Plana, I.; Reinelt, G.; Sanchís Llopis, JM. (2012). New results on the Windy Postman Problem. Mathematical Programming. 132(1-2):309-332. https://doi.org/10.1007/s10107-010-0399-xS3093321321-2Benavent E., Carrotta A., Corberán A., Sanchis J.M., Vigo D.: Lower bounds and heuristics for the windy rural postman problem. Eur. J. Oper. Res. 176, 855–869 (2007)Brucker P. The Chinese postman problem for mixed graphs. In Proceedings of international workshop. Lecture Notes in Computer Science 100, 354–366 (1981)Caprara A., Fischetti M.: $${\{0,\frac{1}{2}\}}$$ -Chvátal-Gomory cuts. Math. Program. 74, 221–235 (1996)Caprara A., Fischetti M., Letchford A.N.: On the separation of maximally violated mod-k cuts. Math. Program. 87, 37–56 (2000)Christof, T., Loebel, A.: PORTA—a polyhedron representation algorithm www.informatik.uni-heidelberg.de/groups/comopt/software/PORTA/ (1998)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An optimal method for the mixed postman problem. In Thoft-Christensen, P. (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences 59, Springer (1984)Corberán A., Plana I., Sanchis J.M.: Zigzag inequalities: a new class of facet-inducing inequalities for arc routing problems. Math. Program. 108, 79–96 (2006)Corberán A., Plana I., Sanchis J.M.: A branch & cut algorithm for the windy general routing problem and special cases. Networks 49, 245–257 (2007)Corberán, A., Plana I., Sanchis, J.M.: Arc routing problems: data instances. www.uv.es/corberan/instancias.htm (2007)Corberán A., Plana I., Sanchis J.M.: The windy general routing polyhedron: a global view of many known arc routing polyhedra. SIAM J. Discrete Math. 22, 606–628 (2008)Grötschel, M., Win, Z.: On the windy postman polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany (1988)Grötschel M., Win Z.: A cutting plane algorithm for the Windy Postman Problem. Math. Program. 55, 339–358 (1992)Guan M.: On the Windy Postman Problem. Discrete Appl. Math. 9, 41–46 (1984)Minieka E.: The Chinese postman problem for mixed networks. Manage. Sci. 25, 643–648 (1979)Naddef D., Rinaldi G.: The symmetric traveling salesman polytope and its graphical relaxation: composition of valid inequalities. Math. Program. 51, 359–400 (1991)Oswald M., Reinelt G., Seitz H.: Applying mod-k cuts for solving linear ordering problems. TOP 17, 158–170 (2009)Papadimitriou C.H.: On the complexity of edge traversing. J. Assoc. Comput. Mach. 23, 544–554 (1976)Ralphs T.K.: On the mixed Chinese postman problem. Oper. Res. Lett. 14, 123–127 (1993)Wenger, K.: Generic Cut Generation Methods for Routing Problems. PhD Dissertation, University of Heidelberg, Germany (2004)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany (1987)Win Z.: On the Windy Postman Problem on eulerian graphs. Math. Program. 44, 97–112 (1989)Zaragoza Martínez F.J.: Series-parallel graphs are windy postman perfect. Discrete Math. 308, 1366–1374 (2008

Min-Max K-vehicles Windy Rural Postman Problem

[EN] In this article the Min-Max version of the windy rural postman problem with several vehicles is introduced. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here an ILP formulation and study its associated polyhedron. Based on its partial description, a branch-and-cut algorithm has been implemented and computational results on a large set of instances are finally presented. (C) 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 54(4),216-226 2009Contract grant sponsor: Ministerio de Education y Ciencia of Spain: Contract gram number: MTM2006-14961-C05-02Benavent López, E.; Corberan, A.; Plana, I.; Sanchís Llopis, JM. (2009). Min-Max K-vehicles Windy Rural Postman Problem. Networks. 54(4):216-226. https://doi.org/10.1002/net.20334S216226544D. Ahr Contributions to multiple postmen problems 2004D. Ahr G. Reinelt “New heuristics and lower bounds for the min-max k -Chinese postman problem” Algorithms-ESA 2002, 10th Annual European Symposium, Rome, Italy, 2002, Lecture Notes in Computer Science 2461 R. Möring R. Raman Springer Berlin 2002 64 74Ahr, D., & Reinelt, G. (2006). A tabu search algorithm for the min–max k-Chinese postman problem. Computers & Operations Research, 33(12), 3403-3422. doi:10.1016/j.cor.2005.02.011D. Applegate R.E. Bixby V. Chvátal W. Cook Finding cuts in the TSP 1995Barahona, F., & Grötschel, M. (1986). On the cycle polytope of a binary matroid. Journal of Combinatorial Theory, Series B, 40(1), 40-62. doi:10.1016/0095-8956(86)90063-8Belenguer, J. M., & Benavent, E. (1998). Computational Optimization and Applications, 10(2), 165-187. doi:10.1023/a:1018316919294Benavent, E., Carrotta, A., Corberán, A., Sanchis, J. M., & Vigo, D. (2007). Lower bounds and heuristics for the Windy Rural Postman Problem. European Journal of Operational Research, 176(2), 855-869. doi:10.1016/j.ejor.2005.09.021N. Christofides V. Campos A. Corberán E. Mota An algorithm for the rural postman problem 1981Christofides, N., Campos, V., Corberán, A., & Mota, E. (1986). An algorithm for the Rural Postman problem on a directed graph. Netflow at Pisa, 155-166. doi:10.1007/bfb0121091Corberán, A., Plana, I., & Sanchis, J. M. (2008). The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics, 22(2), 606-628. doi:10.1137/050640886Corberán, A., Plana, I., & Sanchis, J. M. (2007). A branch & cut algorithm for the windy general routing problem and special cases. Networks, 49(4), 245-257. doi:10.1002/net.20176Eiselt, H. A., Gendreau, M., & Laporte, G. (1995). Arc Routing Problems, Part II: The Rural Postman Problem. Operations Research, 43(3), 399-414. doi:10.1287/opre.43.3.399Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017G. Ghiani D. Laganá G. Laporte R. Musmanno A branch-and-cut algorithm for the undirected capacitated arc routing problem 2007Ghiani, G., & Laporte, G. (2000). A branch-and-cut algorithm for the Undirected Rural Postman Problem. Mathematical Programming, 87(3), 467-481. doi:10.1007/s101070050007Golden, B. L., & Wong, R. T. (1981). Capacitated arc routing problems. Networks, 11(3), 305-315. doi:10.1002/net.3230110308Padberg, M. W., & Rao, M. R. (1982). Odd Minimum Cut-Sets andb-Matchings. Mathematics of Operations Research, 7(1), 67-80. doi:10.1287/moor.7.1.67Pearn, W. L. (1994). Solvable cases of the k-person Chinese postman problem. Operations Research Letters, 16(4), 241-244. doi:10.1016/0167-6377(94)90073-

Self-Assembly of DNA Graphs and Postman Tours

DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and the Windy Postman Problem

Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems

We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general

Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems

[EN] In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.The authors wish to thank the Ministerio de Ciencia y Tecnología of Spain (project TIC2003-05982-C05-01) and the Generalitat Valenciana (Ref: GRUPOS03/189) their support.Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2006). Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems. Mathematical Programming. 108(1):79-96. https://doi.org/10.1007/s10107-005-0643-yS79961081Benavent, E., Carrotta, A., Corberán, A., Sanchis, J.M., Vigo, D.: Lower Bounds and Heuristics for the Windy Rural Postman Problem. Technical Report TR03-2003. Department of Statistics and OR, University of Valencia (Spain). Submitted to EJOR 2003Benavent, E., Corberán, A., Piñana, E., Plana, I., Sanchis, J.M.: New Heuristics for the Windy Rural Postman Problem. To appear in Comput. Oper. Res. 2005Chopra, S., Rinaldi, G.: The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities. SIAM J. Discrete Math. 9 (4), 602–624 (1996)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An Optimal Method for the Mixed Postman Problem. In: P. Thoft-Christensen (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, 59. Berlin: Springer-Verlag 1984Christofides, N., Campos, V., Corberán, A., Mota, E.: An Algorithm for the Rural Postman Problem. Report IC.OR. 81.5. Imperial College, London 1981Corberán, A., Mejía, G., Sanchis, J.M.: New Results on the Mixed General Routing Problem. To appear in Oper. Res. 2005Corberán, A., Mota, E., Sanchis, J.M.: A Comparison of Two Different Formulations for Arc Routing Problems on Mixed Graphs. To appear in Comput. Oper. Res. 2005Corberán, A., Plana, I., Sanchis, J.M.: On the Windy General Routing Polyhedron. In preparation 2005Corberán, A., Romero, A., Sanchis, J.M.: The Mixed General Routing Problem Polyhedron. Math. Programming 96, 103–137 (2003)Cornuèjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Math. Programming 33, 1–27 (1985)Eiselt, H.A., Gendreau, M., Laporte, G.: Arc-Routing Problems, Part 2: the Rural Postman Problem. Oper. Res. 43, 399–414 (1995)Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton, NJ 1962Grötschel, M., Win, Z.: On the Windy Postman Polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany 1988Grötschel, M., Win, Z.: A Cutting Plane Algorithm for the Windy Postman Problem. Math. Programming 55, 339–358 (1992)Guan, M.: On the Windy Postman Problem. Discrete Applied Mathematics 9, 41–46 (1984)Letchford, A.: New inequalities for the General Routing Problem. Eur. J. Oper. Res. 96, 317–322 (1997)Minieka, E.: The Chinese Postman Problem for Mixed Networks. Management Sci. 25, 643–648 (1979)Naddef, D., Rinaldi, G.: The Symmetric Traveling Salesman Polytope and its Graphical Relaxation: Composition of Valid Inequalities. Math. Programming 51, 359–400 (1991)Nobert, Y., Picard, J.C.: An Optimal Algorithm for the Mixed Chinese Postman Problem. Networks 27, 95–108 (1996)Ralphs, T.K.: On the Mixed Chinese Postman Problem. Oper. Res. Lett. 14, 123–127 (1993)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany 198

Arc routing problems: A review of the past, present, and future

[EN] Arc routing problems (ARPs) are defined and introduced. Following a brief history of developments in this area of research, different types of ARPs are described that are currently relevant for study. In addition, particular features of ARPs that are important from a theoretical or practical point of view are discussed. A section on applications describes some of the changes that have occurred from early applications of ARP models to the present day and points the way to emerging topics for study. A final section provides information on libraries and instance repositories for ARPs. The review concludes with some perspectives on future research developments and opportunities for emerging applicationsThis research was supported by the Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional, Grant/Award Number: PGC2018-099428-B-I00. The Research Council of Norway, Grant/Award Numbers: 246825/O70 (DynamITe), 263031/O70 (AXIOM).Corberán, Á.; Eglese, R.; Hasle, G.; Plana, I.; Sanchís Llopis, JM. (2021). Arc routing problems: A review of the past, present, and future. Networks. 77(1):88-115. https://doi.org/10.1002/net.21965S8811577

An updated annotated bibliography on arc routing problems

The number of arc routing publications has increased significantly in the last decade. Such an increase justifies a second annotated bibliography, a sequel to Corberán and Prins (Networks 56 (2010), 50–69), discussing arc routing studies from 2010 onwards. These studies are grouped into three main sections: single vehicle problems, multiple vehicle problems and applications. Each main section catalogs problems according to their specifics. Section 2 is therefore composed of four subsections, namely: the Chinese Postman Problem, the Rural Postman Problem, the General Routing Problem (GRP) and Arc Routing Problems (ARPs) with profits. Section 3, devoted to the multiple vehicle case, begins with three subsections on the Capacitated Arc Routing Problem (CARP) and then delves into several variants of multiple ARPs, ending with GRPs and problems with profits. Section 4 is devoted to applications, including distribution and collection routes, outdoor activities, post-disaster operations, road cleaning and marking. As new applications emerge and existing applications continue to be used and adapted, the future of arc routing research looks promising.info:eu-repo/semantics/publishedVersio