The Mixed Chinese Postman Problem Parameterized by Pathwidth and Treedepth

In the mixed Chinese postman problem (MCPP), given a weighted mixed graph G (it may have both edges and arcs), our aim is to find a closed walk of minimum weight traversing each edge and arc at least once. The MCPP parameterized by the number of edges in G or the number of arcs in G is fixed-parameter tractable as proved by van Bevern et al. in 2014 and Gutin, Jones, and Sheng in 2014, respectively. Solving an open question of van Bevern et al., we show that somewhat unexpectedly the MCPP parameterized by the (undirected) treewidth of G is W[1]-hard. In fact, we prove that even the unweighted MCPP parameterized by the pathwidth of G is W[1]-hard. On the positive side, we show that MCPP parameterized by treedepth is fixed-parameter tractable (even with arbitrary integer weights). We are unaware of any widely studied graph parameters between pathwidth and treedepth and so our results provide a close characterization of the complexity of MCPP.