Structural Parameterizations of the Mixed Chinese Postman Problem

In the Mixed Chinese Postman Problem (MCPP), given a weighted mixed graph G (G may have both edges and arcs), our aim is to find a minimum weight closed walk 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. (2014) and Gutin, Jones and Sheng (2014), respectively. In this paper, we consider the unweighted version of MCPP. Solving an open question of van Bevern et al. (2014), we show that somewhat unexpectedly MCPP parameterized by the (undirected) treewidth of G is W[1]-hard. In fact, we prove that even the 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. We are unaware of any natural graph parameters between pathwidth and treedepth and so our results provide a dichotomy of the complexity of MCPP.

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