= 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>.
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