Separable convex optimization over indegree polytopes
We study separable discrete convex optimization over indegree vectors of orientations of an undirected graph, focusing on the effect of imposing acyclicity. Since the extreme points of the indegree polytope are precisely the indegree vectors of acyclic orientations, this amounts to comparing optimization over its integer points with optimization over its extreme points. Over all orientations, minimizing a vertex-separable discrete convex objective is strongly polynomial-time solvable by minimum-cost flow. Over acyclic orientations, for every fixed discrete convex function shared by all vertices, we obtain a dichotomy: if the function is linear on the positive integers, the problem is solvable in linear time; otherwise exact optimization is NP-hard on connected simple graphs, and APX-hard when the function is non-negative. We also give strongly polynomial-time constant-factor approximation algorithms for broad classes of convex and concave objectives. For the $p$-power sum of indegrees, this yields a $2^p$-approximation for every fixed integer $p\geq1$, while the problem is APX-complete for every fixed integer $p\geq2$; if $p$ is part of the input, no deterministic polynomial-time $2^{p^{1-\varepsilon}}$-approximation exists for any fixed $\varepsilon>0$ unless $\mathrm{P}=\mathrm{NP}$. We further give an exact $O(2^{|V|}|V|+\operatorname{poly}(|V|,|E|))$-time dynamic program for general separable objectives, with a matching lower bound in the exponent under the Exponential Time Hypothesis. Finally, decreasingly minimal and increasingly maximal acyclic orientations need not coincide and are both NP-hard, while minimizing the maximum weighted indegree over acyclic orientations remains strongly polynomial-time solvable.