Inclusion-wise extremal knapsack problems are (mostly) hard to approximate

We study two inclusion-wise extremal variants of the 0-1 Knapsack Problem (KP). Given $n$ items with integer weights and nonnegative costs or profits, the Minimum-Cost Maximal Knapsack Packing Problem seeks a minimum-cost packing for capacity $c$ to which no item can be added, while the Maximum-Profit Minimal Knapsack Cover Problem seeks a maximum-profit cover for threshold $d$ from which no item can be removed. Both problems are weakly $\mathrm{NP}$-hard and pseudo-polynomially solvable, as is the classical KP, but their approximability is fundamentally different: unless $\mathrm{P}=\mathrm{NP}$, with positive values neither problem admits a polynomial-time $2^{p(n)}$-approximation for any polynomial $p$; allowing zero values rules out every multiplicative factor. Even with unit costs for packing and unit profits for covering, neither admits an $n^{1-\varepsilon}$-approximation for any fixed $\varepsilon>0$, whereas a trivial $n$-approximation exists. We identify restrictions that restore approximability. If every item weighs at least a fixed fraction of the capacity, respectively of the threshold, both problems can be solved exactly in polynomial time. For the cover problem, if each weight $w_i$ satisfies $w_i-1\geδ(d-1)$, we give an approximation scheme for arbitrary profits whose running time is polynomial in $n$, $1/\varepsilon$, and $1/δ$. If values equal weights, a fully polynomial-time approximation scheme (FPTAS) was already known for the packing problem; we give an FPTAS for the cover problem, improving on the previously known factor $2$. If value-to-weight ratios differ by at most a factor $κ$, we obtain a $(1+\varepsilon)κ$-approximation for both problems.