Finding postal carrier walk paths in mixed graphs

The postman problem requires finding a lowest cost tour in a connected graph that traverses each edge at least once. In this paper we first give a brief survey of the literature on postman problems including the original Chinese postman problem on undirected graphs, the windy Chinese postman problem on graphs where the cost of an arc depends on the direction the arc is transversed, the directed postman problem on graphs with directed edges, and the mixed postman problem on graphs in which there are some directed and some undirected arcs. We show how the mixed postman problem can be solved as an integer program, using the formulation of Gendreau, Laporte and Zhao, by a new row addition branch and bound algorithm, which is a modification of the column subtraction algorithm for set partitioning problems of Harche and Thompson. Computational experience shows that a ‘slack variable’ heuristic is very effective in finding good solutions that are frequently optimal for these problems.