Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs

A Model-Based Heuristic to the Min Max K-Arc Routing for Connectivity Problem

We consider the post-disaster road clearing problem with the goal of restoring network connectivity in shortest time. Given a set of blocked edges in the road network, teams positioned at depot nodes are dispatched to open a subset of them that reconnects the network. After a team finishes working on an edge, others can traverse it. The problem is to find coordinated routes for the teams. We generate a feasible solution using a constructive heuristic algorithm after solving a relaxed mixed integer program. In almost 70 percent of the instances generated both randomly and from Istanbul data, the relaxation solution turned out to be feasible, i.e. optimal for the original problem

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

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-

Solving, Generating, and Modeling Arc Routing Problems

Arc routing problems are an important class of network optimization problems. In this dissertation, we develop an open source library with solvers that can be applied to several uncapacitated arc routing problems. The library has a flexible architecture and the ability to visualize real-world street networks. We also develop a software tool that allows users to generate arc routing instances directly from an open source map database. Our tool has a visualization capability that can produce images of routes overlaid on a specific instance. We model and solve two variants of the standard arc routing problem: (1) the windy rural postman problem with zigzag time windows and (2) the min-max K windy rural postman problem. In the first variant, we allow servicing of both sides of some streets in a network, that is, a vehicle can service a street by zigzagging. We combine insertion and local search techniques to produce high-quality solutions to a set of test instances. In the second variant, we design a cluster-first, route-second heuristic that compares favorably to an existing heuristic and produces routes that are intuitively appealing. Finally, we show how to partition a street network into routes that are compact, balanced, and visually appealing

A Branch-Price-and-Cut Algorithm for the Min-Max k -Vehicle Windy Rural Postman Problem

[EN] The min-max k -vehicles windy rural postman problem consists of minimizing the maximal distance traveled by a vehicle to find a set of balanced routes that jointly service all the required edges in a windy graph. This is a very difficult problem, for which a branch-and-cut algorithm has already been proposed, providing good results when the number of vehicles is small. In this article, we present a branch-price-and-cut method capable of obtaining optimal solutions for this problem when the number of vehicles is larger for the same set of required edges. Extensive computational results on instances from the literature are presented.Contract grant sponsor: Ministerio de Education y Ciencia of Spain: Contract gram number: MTM2006-14961-C05-02 Canadian Natural Sciences and Engineering Research Council; Contract grant number: 157935-07Benavent Lopez, E.; Corberán, A.; Desaulniers, G.; Lessard, F.; Plana, I.; Sanchís Llopis, JM. (2014). A Branch-Price-and-Cut Algorithm for the Min-Max k -Vehicle Windy Rural Postman Problem. Networks. 63(1):34-45. https://doi.org/10.1002/net.21520S3445631Baldacci, R., Mingozzi, A., & Roberti, R. (2011). New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem. Operations Research, 59(5), 1269-1283. doi:10.1287/opre.1110.0975Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., & Vance, P. H. (1998). Branch-and-Price: Column Generation for Solving Huge Integer Programs. Operations Research, 46(3), 316-329. doi:10.1287/opre.46.3.316Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2009). Min-MaxK-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/s10287-009-0119-2Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2011). New facets and an enhanced branch-and-cut for the min-max K-vehicles windy rural postman problem. Networks, 58(4), 255-272. doi:10.1002/net.20469Boland, N., Dethridge, J., & Dumitrescu, I. (2006). Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Operations Research Letters, 34(1), 58-68. doi:10.1016/j.orl.2004.11.011Corberán, 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:10.1137/050640886Á. Corberán I. Plana J.M. Sanchis Arc routing problems: Data instances www.uv.es/corberan/instancias.htm 2007Dantzig, G. B., & Wolfe, P. (1960). Decomposition Principle for Linear Programs. Operations Research, 8(1), 101-111. doi:10.1287/opre.8.1.101Desaulniers, G., Desrosiers, J., & Spoorendonk, S. (2011). Cutting planes for branch-and-price algorithms. Networks, 58(4), 301-310. doi:10.1002/net.20471Desaulniers, G., Lessard, F., & Hadjar, A. (2008). Tabu Search, Partial Elementarity, and Generalizedk-Path Inequalities for the Vehicle Routing Problem with Time Windows. Transportation Science, 42(3), 387-404. doi:10.1287/trsc.1070.0223Dror, M. (1994). Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW. Operations Research, 42(5), 977-978. doi:10.1287/opre.42.5.977Gilmore, P. C., & Gomory, R. E. (1961). A Linear Programming Approach to the Cutting-Stock Problem. Operations Research, 9(6), 849-859. doi:10.1287/opre.9.6.849Hadjar, A., Marcotte, O., & Soumis, F. (2006). A Branch-and-Cut Algorithm for the Multiple Depot Vehicle Scheduling Problem. Operations Research, 54(1), 130-149. doi:10.1287/opre.1050.0240Hoffman, K. L., & Padberg, M. (1993). Solving Airline Crew Scheduling Problems by Branch-and-Cut. Management Science, 39(6), 657-682. doi:10.1287/mnsc.39.6.657Jepsen, M., Petersen, B., Spoorendonk, S., & Pisinger, D. (2008). Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows. Operations Research, 56(2), 497-511. doi:10.1287/opre.1070.0449Lübbecke, M. E., & Desrosiers, J. (2005). Selected Topics in Column Generation. Operations Research, 53(6), 1007-1023. doi:10.1287/opre.1050.0234Padberg, 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)90073-6Righini, G., & Salani, M. (2006). Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optimization, 3(3), 255-273. doi:10.1016/j.disopt.2006.05.007Righini, G., & Salani, M. (2008). New dynamic programming algorithms for the resource constrained elementary shortest path problem. Networks, 51(3), 155-170. doi:10.1002/net.20212Ropke, S., & Cordeau, J.-F. (2009). Branch and Cut and Price for the Pickup and Delivery Problem with Time Windows. Transportation Science, 43(3), 267-286. doi:10.1287/trsc.1090.027