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

Search for an Immobile Hider on a Stochastic Network

Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.Comment: 28 pages, 9 figure

Efficient routing of snow removal vehicles

This research addresses the problem of finding a minimum cost set of routes for vehicles in a road network subject to some constraints. Extensions, such as multiple service requirements, and mixed networks have been considered. Variations of this problem exist in many practical applications such as snow removal, refuse collection, mail delivery, etc. An exact algorithm was developed using integer programming to solve small size problems. Since the problem is NP-hard, a heuristic algorithm needs to be developed. An algorithm was developed based on the Greedy Randomized Adaptive Search Procedure (GRASP) heuristic, in which each replication consists of applying a construction heuristic to find feasible and good quality solutions, followed by a local search heuristic. A simulated annealing heuristic was developed to improve the solutions obtained from the construction heuristic. The best overall solution was selected from the results of several replications. The heuristic was tested on four sets of problem instances (total of 115 instances) obtained from the literature. The simulated annealing heuristic was able to achieve average improvements of up to 26.36% over the construction results on these problem instances. The results obtained with the developed heuristic were compared to the results obtained with recent heuristics developed by other authors. The developed heuristic improved the best-known solution found by other authors on 18 of the 115 instances and matched the results on 89 of those instances. It worked specially better with larger problems. The average deviations to known lower bounds for all four datasets were found to range between 0.21 and 2.61%

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

Automatic March tests generation for static and dynamic faults in SRAMs

New memory production modern technologies introduce new classes of faults usually referred to as dynamic memory faults. Although some hand-made March tests to deal with these new faults have been published, the problem of automatically generate March tests for dynamic faults has still to be addressed, in this paper we propose a new approach to automatically generate March tests with minimal length for both static and dynamic faults. The proposed approach resorts to a formal model to represent faulty behaviors in a memory and to simplify the generation of the corresponding tests

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

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

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