Distinct optimal solutions for polynomial indegree objectives

Prove or disprove that, for every integer exponent \(c\ge2\), minimizing \(\sum_{v\in V}\varrho(v)^c\) over acyclic orientations yields a set of optimal orientations distinct from that obtained for every other exponent.

Background

The paper observes that the optimal acyclic orientations for the quadratic and cubic objectives φ(z)=z2\varphi(z)=z^2 and φ(z)=z3\varphi(z)=z^3 do not generally coincide. It also gives an example showing that a cubic optimum need not be a quadratic optimum.

The authors conjecture a general divergence phenomenon for all integer polynomial exponents at least two, but do not prove it.

References

For instance, the optimal acyclic orientations for $\varphi(z) = z2$ and for $\varphi(z) = z3$ do not coincide in general, and we conjecture that for every integer exponent $c \geq 2$, the problems for $\varphi(z) = zc$ admit distinct sets of optimal solutions.

— Separable convex optimization over indegree polytopes  (2509.06182 - Borsik et al., 7 Sep 2025) in Remark 3.1 in Section 3 (Remark~\ref{sec:minSumH})