Simple-graph complexity of strictly convex indegree minimization

Determine the computational complexity, for simple graphs, of minimizing the sum of a discrete strictly convex function of the left-degrees over all vertex orderings, equivalently minimizing the corresponding separable strictly convex indegree objective over acyclic orientations.

Background

The paper proves that minimizing $\sum_{v\in V}\varphi(\cev d(v))$ is NP-hard for every discrete strictly convex function $\varphi$ when parallel edges are permitted. The authors also prove NP-hardness for the square objective $\varphi(z)=z^2$ on simple graphs, but their general hardness result does not extend to arbitrary discrete strictly convex functions on simple graphs.

Consequently, it remains unresolved whether the full strictly convex optimization problem is computationally tractable or NP-hard when the input graph is required to be simple.