Approximation complexity for fixed value-to-weight spread

Determine whether MinMax-Pack and MaxMin-Cov with a fixed value-to-weight spread κ>1 admit a polynomial-time approximation scheme, or prove that these fixed-spread problem classes are APX-hard.

Background

For spread κ=1, corresponding to values proportional to weights, both problems admit fully polynomial-time approximation schemes. For unbounded spread, the paper proves extremely strong inapproximability. The intermediate regime of a fixed constant κ>1 is unresolved: the existing (1+ε)κ-approximation does not establish a PTAS, and no APX-hardness result is given. The authors also note the related strongly correlated objective class v_i=w_i+const.

References

For a fixed $\kappa>1$, is there a PTAS, or are the problems $APX$-hard?

— Inclusion-wise extremal knapsack problems are (mostly) hard to approximate  (2610.18017 - Ciccarelli et al., 13 Oct 2026) in Section 6, paragraph “Open problems,” item 2