No-waste allocation

For nonnegative additive valuations, the requirement that each allocated item be assigned to an agent who values it strictly positively. This condition alone does not require allocating every item.

Let $S_i=\{j:v_{ij}>0\}$ and restrict agent $i$'s constraint domain to subsets of $S_i$, or to nonempty subsets when required. If $S_i\cap S_{\ell }=\varnothing$, the two bundles are automatically disjoint and each agent values the other's bundle at zero. Both directional envy-free allocation binary constraints are then automatic. Thus positive-support overlaps determine which pairs can still matter in the residual constraint graph.