Arc Routing pp 171-196 | Cite as

# Chinese Postman and Euler Tour Problems in Bi-Directed Graphs

## Abstract

Euler tours occupy an interesting position in the history of graph theory. Current interest in this area is due to problems involving tours where service is required along arcs of the tour rather than at nodes. Examples of problems of this type involve mail delivery, snow removal, street cleaning, trash pickup, etc. In considering such problems involving city streets, the nodes are intersections and the arcs are roadways between intersections. However, there are one-way streets and two-way streets. If the service must be done on both sides of a two-way street then two one-way streets (one in each direction) can replace it. However, there are situations where the street, even though two-way, need only be traversed once. For example, mail delivery in a more rural or suburban setting may require traversing the street or road only once. In a rural road, mail delivery is frequently done on one side of the road, and those who live on the other side must cross the road to get their mail. In setting up the routes, though, the road could be traversed in either direction. Thus, we consider graphs with two types of connections: directed and undirected. Other postman problems [14] have been considered.

## Keywords

Span Tree Dual Graph Euler Tour Mixed Graph Bipartite Subgraph## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Barnhart, C., N. Boland, L. Clarke, E.L. Johnson, G.L. Nemhauser, R. Shenoi, “Flight String Models for Aircraft Fleeting and Routing”
*Transportation Science*Vol. 32, No. 3, pp. 208–220, August 1998.MATHCrossRefGoogle Scholar - [2]Clarke, L., E.L. Johnson, G.L. Nemhauser, J. Zhu, “The Aircraft Rotation Problem”,
*Annals of Operations Research 69*, pp. 33–46, 1996.CrossRefGoogle Scholar - [3]M. Conforti, G. Cornujols, A. Kapoor and K. Vuskovic, “Perfect, Ideal and Balanced Matrices”, (1996).Google Scholar
- [4]Cornuéjols, G. “Combinatorial Optimization: Packing and Covering”, Lecture Notes, Carrnegie Mellon University, May 1999.Google Scholar
- [5]Cornuéjols, G. and B. Guenin, “On Ideal Binary Clutters and a Conjecture of Seymour”, in preparation.Google Scholar
- [6]Edmonds, J. “The Chinese Postman Problem”,
*Operations Research*13, Suppl. 1, pp. 373, 1965.Google Scholar - [7]Edmonds, J. and E.L. Johnson, “Matching: A Well-Solved Class of Integer Linear Programs”,
*Combinatorial Structures and Their Applications*, proceedings from Calgary, Alberta, Canada, June 1969, pp. 89–92.Google Scholar - [8]Edmonds, J. and R.M. Karp, “Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems”,
*Combinatorial Structures and Their Applications*, proceedings from Calgary, Alberta, Canada, June 1969, pp. 93–96.Google Scholar - [9]Eiselt, H.A. and G. Laporte, “A Historical Perspective on Arc Routing”, this volume.Google Scholar
- [10]Ford, L.R. and D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., 1962.MATHGoogle Scholar
- [11]Fulkerson, D.R. “Blocking Polyhedra,” in B. Harris, ed.,
*Graph Theory and its Applications*(Academic Press, NY), pp. 93–112, 1970.Google Scholar - [12]Gomory, R.E. “Some Polyhedra Related to Combinatorial Problems,”
*Linear Algebra and it Applications*2, pp. 451–558, 1969.MathSciNetMATHCrossRefGoogle Scholar - [13]Grötschel, M. and W.R. Pulleyblank, “Weakly Bipartite Graphs”,
*Operations Research Letters 1*pp. 23–27, 1981.MathSciNetMATHCrossRefGoogle Scholar - [14]Grötschel, M. and Z. Win, “A Cutting Plane Algorithm for the windy Postman Problem”,
*Mathematical Programming*55 (1992), pp. 339–358.MathSciNetMATHCrossRefGoogle Scholar - [15]Guan, M. “Graphic Programming Using Odd or Even Points”,
*Chinese Mathematics*1 (1962), pp. 273–377.Google Scholar - [16]Hane, C., C. Barnhart, E.L. Johnson, R. Marsten, G.L. Nemhauser, G. Sigismondi, “The Fleet Assignment Problem: Solving a Large Integer Program”,
*Mathematical Programming*70, pp. 211–232, 1995.MathSciNetMATHGoogle Scholar - [17]Johnson, E.L. “On Binary Group Problems having the Fulkerson Property”,
*Combinatorial Optimization*B. Simeone (ed.), Springer-Verlag, Berlin and Heidelberg, pp. 57–112, 1989.CrossRefGoogle Scholar - [18]Johnson, E.L. and J. Edmonds, “Matchings, Euler Tours and the Chinese Postman”,
*Mathematical Programming*, Vol. 5, pp. 88–124, 1973.MathSciNetMATHCrossRefGoogle Scholar - [19]Johnson, E.L. and G. Gastou, “Binary Group and Chinese Postman Polyhedra”,
*Mathematical Programming*, Vol. 5, pp. 88–124, 1973.MathSciNetMATHCrossRefGoogle Scholar - [20]Johnson, E.L. and S. Mosterts, “On Four Problems in Graph Theory,”
*SIAM Journal on Algebraic and Discrete Methods*, Vol. 2, pp. 163–185, 1987.MathSciNetCrossRefGoogle Scholar - [21]Khachian, L.G. “A Polynomial Algorithm in Linear Programming”,
*Soviet Mathematics Doklady*20 pp. 191–194, 1979.Google Scholar - [22]Lehman, A. “On the Width-Length Inequality”,
*Mathematical Programming*17(1979), pp. 403–417.MathSciNetMATHCrossRefGoogle Scholar - [23]Nobert, Y. and J.-C. Picard, “An Optimal Algorithm for the Mixed Chinese Postman Problem”,
*Networks*Vol. 27 (1996), pp. 95–108.MathSciNetMATHCrossRefGoogle Scholar - [24]Padberg, M. and M. Rao, “Odd Minimum Cut-Sets and B-Matchings”,
*Math. of Operations Research*, Vol. 7 No. 1 (1982), pp. 67–80.MathSciNetMATHCrossRefGoogle Scholar - [25]Roberts, F.S. and J. Spencer, “A Characterization of Clique Graphs”,
*Combinatorial Structures and Their Applications*, proceedings from Calgary, Alberta, Canada, June 1969, pp. 367–368.Google Scholar - [26]Seymour, P.D. “Matroids with the Max-Flow Min-Cut Property”,
*Journal of Combinatorial Theory Series B*23 (1977), pp. 189–222.MathSciNetMATHCrossRefGoogle Scholar - [27]Stone, A.H. “Some Combinatorial Problems in General Topology”,
*Combinatorial Structures and Their Applications*, proceedings from Calgary, Alberta, Canada, June 1969, pp. 413–416.Google Scholar - [28]Tutte, W.T. “Lectures on Matroids”,
*Journal of Research of the National Bureau of Standards Section B*69 pp. 1–47, 1965.MathSciNetMATHCrossRefGoogle Scholar