Inverse knapsack at two capacities: which pairs of value-cardinality hulls are realizable?

Abstract

One item set evaluated at two capacities $R<D$ produces two concave hulls of optimal value against cardinality. We ask which prescribed pairs arise. Exchange arguments give a necessary system on vertex witnesses, exchange closure (EC), whose scalar consequences form the linear closure. We exhibit a pair satisfying every scalar test that fails EC, so the linear closure is strictly larger already at larger terminal count three; and a globally coherent EC witness admitting no common-size representation although its target pair is realizable. Under a cardinality cap, a four-band classification of one family gives exact thresholds for cap-four realizability, uncapped realizability and the vertex-only capped closure. Pairs whose larger terminal count is at most two are characterized. Under the explicit encoding the decision problem lies in $Σ_2^p$ and is polynomial-time for fixed terminal cardinalities. Exact finite certificates establish agreement of the scalar and witness conditions on six specified domains; sufficiency of uncapped EC remains open.