Arc Routing: Complexity and Approximability

The majority of arc routing problems can be viewed as variants of the classical Chinese Postman Problem (CPP). Restating the generic problem, let G = (V,E) be a connected graph (undirected) with V a finite set (the nodes) and E ⊂ V × V be the set of edges. In addition, we have a real valued weight (distance) W ij ≥ 0, ∀(i, j) ∈E, and a design problem: “Construct a least distance traversal sequence of all the edges in E starting at and returning to the same node.” This is in essence the Chinese Postman Problem as posed by Meigu Guan (Mei-Ko Kwan) in 1962, in the Chinese Mathematics journal which is the main reason why we refer to this problem as the CPP. The historical overview of arc routing and variants of CPP are eloquently described by Eiselt and Laporte (this book), however we examine the Guan (1962) work for its illustration of the computational aspects when solving the CPP and related problems. As pointed out in Edmonds and Johnson (1973), the CPP can be separated into two parts: given an arbitrary (connected) graph G, duplicate a set of edges in E of minimal total weight to transform G into Ĝ (an even degree graph) which admits an Euler tour (a closed tour which traverses exactly once every edge in the graph), and then construct an Euler tour on Ĝ.

[1]  Paul Abraham Mullaseril.  Capacitated rural postman problem with time windows and split delivery , 1997 .

[2]  H. Fleischner.  Traversing Graphs: The Eulerian and Hamiltonian Theme , 2000 .

[3]  J. Picard,et al.  An optimal algorithm for the mixed Chinese postman problem , 1996 .

[4]  Ulrich Derigs,et al.  An analysis of alternative strategies for implementing matching algorithms , 1983, Networks.

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  Ted K. Ralphs,et al.  On the mixed Chinese postman problem , 1993, Oper. Res. Lett..

[7]  Carsten Lund,et al.  Hardness of approximations , 1996 .

[8]  William Pulleyblank,et al.  Eulerian Orientations and Circulations , 1985 .

[9]  C. S. Orloff.  A fundamental problem in vehicle routing , 1974, Networks.

[10]  P. Boas.  Machine models and simulations , 1991 .

[11]  Pierce point minimization and optimal torch path determination in flame cutting , 1984 .

[12]  Gilbert Laporte,et al.  Capacitated Vehicle Routing on Trees , 1991, Oper. Res..

[13]  Michael O. Ball,et al.  Sequencing of Insertions in Printed Circuit Board Assembly , 1988, Oper. Res..

[14]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[15]  Enrique Benavent,et al.  Analisis de heuristicos para el problema del cartero rural , 1985 .

[16]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[17]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[18]  Gary J. Koehler,et al.  The mixed postman problem , 1979, Discret. Appl. Math..

[19]  B. Gavish,et al.  Heuristics for unequal weight delivery problems with a fixed error guarantee , 1987 .

[20]  Lars Døvling Andersen,et al.  The NP-completeness of finding A-trails in Eulerian graphs and of finding spanning trees in hypergraphs , 1995 .

[21]  P. A. Mullaseril,et al.  Livestock Feed Distribution and Arc Traversal Problems , 2000 .

[22]  André Langevin,et al.  Transformations and Exact Node Routing Solutions by Column Generation , 2000 .

[23]  Michel Gendreau,et al.  Arc Routing Problems, Part II: The Rural Postman Problem , 1995, Oper. Res..

[24]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[25]  Sanjeev Arora.  Probabilistic checking of proofs and hardness of approximation problems , 1995 .

[26]  Moshe Dror,et al.  Generalized Steiner Problems and Other Variants , 2000, J. Comb. Optim..

[27]  Moshe Dror,et al.  Routing electric meter readers , 1979, Comput. Oper. Res..

[28]  Jack Edmonds,et al.  Matching, Euler tours and the Chinese postman , 1973, Math. Program..

[29]  J. C. Bean,et al.  An efficient transformation of the generalized traveling salesman problem , 1993 .

[30]  Ángel Corberán,et al.  Linear Programming Based Methods for Solving Arc Routing Problems , 2000 .

[31]  Klaus Jansen.  Bounds for the general capacitated routing problem , 1993, Networks.

[32]  Gilbert Laporte,et al.  Arc Routing Problems, Part I: The Chinese Postman Problem , 1995, Oper. Res..

[33]  C. S. Orloff.  On general routing problems: Comments , 1976, Networks.

[34]  Leen Stougie,et al.  ANALYSIS OF HEURISTICS FOR VEHICLE ROUTING PROBLEMS , 1988 .

[35]  E. Minieka.  The Chinese Postman Problem for Mixed Networks , 1979 .

[36]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[37]  Á. Corberán,et al.  A polyhedral approach to the rural postman problem , 1994 .

[38]  Gilbert Laporte.  Modeling and solving several classes of arc routing problems as traveling salesman problems , 1997, Comput. Oper. Res..

[39]  Moshe Dror,et al.  Postman tour on a graph with precedence relation on arcs , 1987, Networks.

[40]  Jan van Leeuwen,et al.  Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity , 1994 .

[41]  Jan Karel Lenstra,et al.  On general routing problems , 1976, Networks.

[42]  Bruce L. Golden,et al.  Capacitated arc routing problems , 1981, Networks.

[43]  Balaji Raghavachari,et al.  A 3/2-Approximation Algorithm for the Mixed Postman Problem , 1999, SIAM J. Discret. Math..

[44]  José-Manuel Belenguer,et al.  The Capacitated Arc Routing Problem: Valid Inequalities and Facets , 1998, Comput. Optim. Appl..

[45]  F. Harary,et al.  On Eulerian and Hamiltonian Graphs and Line Graphs , 1965, Canadian Mathematical Bulletin.

[46]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[47]  Moshe Dror,et al.  Combinatorial Optimization in a Cattle Yard: Feed Distribution, Vehicle Scheduling, Lot Sizing, and Dynamic Pen Assignment , 1998 .

[48]  Gianpaolo Ghiani,et al.  An algorithm for the hierarchical Chinese postman problem , 2000, Oper. Res. Lett..

[49]  Chul E. Kim,et al.  Approximation Algorithms for Some Routing Problems , 1978, SIAM J. Comput..

[50]  Udi Manber,et al.  On non-intersecting Eulerian circuits , 1987, Discret. Appl. Math..

[51]  Greg N. Frederickson,et al.  Approximation Algorithms for Some Postman Problems , 1979, JACM.

[52]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[53]  Balaji Raghavachari,et al.  Approximation Algorithms for the Mixed Postman Problem , 1998, IPCO.

[54]  Balaji Raghavachari,et al.  Approximation algorithms for the asymmetric postman problem , 1999, SODA '99.

[55]  Moshe Dror,et al.  A Generalized Traveling Salesman Problem Approach to the Directed Clustered Rural Postman Problem , 1997, Transp. Sci..

[56]  Klaus Jansen,et al.  An Approximation Algorithm for the General Routing Problem , 1992, Inf. Process. Lett..

[57]  Adam N. Letchford,et al.  Polyhedral Theory for Arc Routing Problems , 2000 .

[58]  Zaw Win,et al.  On the Windy Postman Problem on eulerian graphs , 1989, Math. Program..

[59]  Meigu Guan,et al.  On the windy postman problem , 1984, Discret. Appl. Math..

[60]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[61]  Zvi Galil,et al.  Cyclic Ordering is NP-Complete , 1977, Theor. Comput. Sci..

[62]  Bruce L. Golden,et al.  VEHICLE ROUTING: METHODS AND STUDIES , 1988 .

[63]  Gilbert Laporte,et al.  A Historical Perspective on Arc Routing , 2000 .

[64]  Christos H. Papadimitriou,et al.  On the complexity of edge traversing , 1976, J. ACM.