Parameterized complexity of the no-small-items regime

Determine whether MinMax-Pack and MaxMin-Cov restricted to instances with item weights bounded below by a δ-fraction of the capacity or threshold admit an FPT algorithm parameterized by 1/δ, or prove that they are W[1]-hard with respect to this parameter.

Background

When every item has weight at least a fixed δ-fraction of the relevant capacity or threshold, the paper gives an exact algorithm with running time n^{O(1/δ)}. Consequently, the problem is in XP when parameterized by 1/δ, but the parameter appears in the exponent of n rather than solely in a multiplicative parameter-dependent factor. The unresolved issue is whether this dependence can be improved to fixed-parameter tractability or whether W[1]-hardness prevents such an improvement.