Optimal polynomial-time approximation ratio

Determine the optimal polynomial-time approximation ratio for fractional majority coloring.

Background

The paper gives a randomized polynomial-time construction with approximation ratio arbitrarily close to 523/210 < 2.491 and proves inapproximability thresholds below 72/71 multiplicatively and below 3/142 additively.

These upper and lower algorithmic results do not determine the best achievable polynomial-time approximation ratio. The authors explicitly leave this optimization problem unresolved, noting that acyclic oriented digraphs of maximum outdegree two already form a natural restricted class in which to study it.

References

The constructive and hardness results leave open the optimal polynomial-time approximation ratio for fractional majority coloring. The hardness already holds for acyclic oriented digraphs of maximum outdegree two, making this restricted class a natural setting in which to study that question.

— Fractional majority coloring of digraphs: bounds and inapproximability  (2610.11997 - Ben-Ameur et al., 8 Oct 2026) in Section 5, Concluding remarks