On the complexity of the Eulerian closed walk with precedence path constraints problem

The Eulerian closed walk problem in a digraph is a well-known polynomial-time solvable problem. In this paper, we show that if we impose the feasible solutions to fulfill some precedence constraints specified by paths of the digraph, then the problem becomes NP-complete. We also present a polynomial-time algorithm to solve this variant of the Eulerian closed walk problem when the paths are arc-disjoint. We also give necessary and sufficient conditions for the existence of feasible solutions in this polynomial-time solvable case.