Abstract.
This paper studies a class of delivery problems associated with the Chinese postman problem and a corresponding class of delivery games. A delivery problem in this class is determined by a connected graph, a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office. It is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server, who starts at the post office and is expected to traverse all edges in the graph before returning to the post office. A graph G is called Chinese postman-submodular, or, for short, CP-submodular (CP-totally balanced, CP-balanced, respectively) if for each delivery problem in which G is the underlying graph the associated delivery game is submodular (totally balanced, balanced, respectively).
For undirected graphs we prove that CP-submodular graphs and CP-totally balanced graphs are weakly cyclic graphs and conversely. An undirected graph is shown to be CP-balanced if and only if it is a weakly Euler graph. For directed graphs, CP-submodular graphs can be characterized by directed weakly cyclic graphs. Further, it is proven that any strongly connected directed graph is CP-balanced. For mixed graphs it is shown that a graph is CP-submodular if and only if it is a mixed weakly cyclic graph.
Finally, we note that undirected, directed and mixed weakly cyclic graphs can be recognized in linear time.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received May 20, 1997 / Revised version received August 18, 1998¶Published online June 11, 1999
Rights and permissions
About this article
Cite this article
Granot, D., Hamers, H. & Tijs, S. On some balanced, totally balanced and submodular delivery games. Math. Program. 86, 355–366 (1999). https://doi.org/10.1007/s101070050093
Issue Date:
DOI: https://doi.org/10.1007/s101070050093