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In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given aconnected weighted digraph $G$ and asked to find $k$ non-empty closed directedwalks covering all arcs of $G$ such that the total weight of the walks isminimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128)asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter.We prove that the $k$-DCPP is fixed-parameter tractable.  We also consider a related problem of finding $k$ arc-disjoint directedcycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showedthat this problem is W[1]-hard for general digraphs. Generalizing anotherresult by Slivkins, we prove that the problem is fixed-parameter tractable forEuler digraphs. The corresponding problem on vertex-disjoint cycles in Eulerdigraphs remains W[1]-hard even for Euler digraphs.

Parameterized Directed $k$-Chinese Postman Problem and $k$ Arc-Disjoint Cycles Problem on Euler Digraphs