Traversing a graph with respect to some partial order
This is a job-shop scheduling problem, known as $1|\textrm{prec},p_i=1|\Sigma w_i C_i$. It is $NP$-complete.

Lenstra, J.K. and Rinnooy Kan, A.H.G.: Complexity of Scheduling under Precedence Constraints. In Operations Research, vol. 26, pp. 22-35 (1978)

Sadly the paper is not open-access. They show $NP$-hardness by reduction from Linear Arrangement (also known as Graph Bandwidth)