Uniform constant-factor approximation across all latencies

Establish whether there exists a universal constant δ>0 such that, for every latency L, the Young domination number γ^L can be approximated within a factor δ in polynomial time.

Background

The paper provides a 4-approximation for infinite latency and, for every fixed finite latency L, a constant-factor approximation whose ratio and polynomial running-time exponent depend on L. The authors leave unresolved whether a single approximation factor can be chosen uniformly for all latencies, with a polynomial-time algorithm available for each L.

References

These results suggest the following questions. Is there a constant $\delta>0$ such that for every $L$, there is an algorithm that approximates $\gammaL$ up to factor $\delta$ in polynomial time?

— Young domination on Hamming rectangles  (2501.03788 - Gravner et al., 7 Jan 2025) in Section 1, item 2 of Section 7 ("Open problems and possible further directions")