The Hierarchical Chinese Postman Problem: The slightest disorder makes it hard, yet disconnectedness is manageable

The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges graph respecting precedence constraints given by partial order on classes edges. We show that the special case with connected NP-hard even orders decomposable into chain and an incomparable class. For linearly ordered (possibly disconnected) classes, we get 5/3-approximations fixed-parameter algorithms transferring results from Rural Problem.