Approximation scheme for two-sided minimisation with relative slack

Establish whether minimising v(S) subject to ℓ≤w(S)≤u admits an approximation scheme whose running time is polynomial in 1/δ under the promise u−ℓ≥δu, thereby providing a packing-side analogue of the approximation scheme for the maximisation version.

Background

The paper identifies a two-sided residual optimisation problem on the packing side: minimise the value of a set subject to simultaneous lower and upper bounds on its total weight. Feasibility is NP-hard in general because the special case ℓ=u encodes SUBSET SUM. By contrast, a related maximisation problem admits an approximation scheme whose complexity is polynomial in the inverse relative slack. The unresolved question asks whether an analogous scheme exists for the minimisation problem when the interval width satisfies u−ℓ≥δu.

References

Under the promise $u-\ell\ge\delta u$, is there an approximation scheme polynomial in $1/\delta$, analogous to the scheme of \citet{Xu2013} for the maximisation version?

— 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 3