Simple allocation schemes for two capacity constraints

Develop allocation schemes that use a small constant number of signals for each right-hand-side node and quickly allocate arriving left-hand-side items so that the resulting allocations are approximately feasible for instances with two capacity constraints.

Background

The paper considers bipartite allocation problems in which every right-hand-side node has two unrelated capacity constraints, exemplified by weight and volume limits at warehouses, while each left-hand-side item consumes both types of capacity. Although a perfect matching may satisfy both constraints, the paper shows that ordinary proportional allocation can violate one of the constraints by an arbitrarily large factor, even on complete bipartite graphs.

The unresolved issue is whether an alternative simple allocation scheme can use only a small constant number of signals per right-hand-side node and still produce allocations that are approximately feasible for all instances with two capacity constraints.

References

It remains open if one can find other `simple' allocation schemes that can use a small constant number of signals for each right-hand side node that can be used to quickly allocate arriving items from the left-hand side so that the resulting allocations are (approximately) feasible for instances with two capacity constraints.

— Perfect Fractional Matchings in Bipartite Graphs Via Proportional Allocations  (2510.01107 - Hathcock et al., 1 Oct 2025) in Section 4, “Proportional Allocations for Two Capacity Constraints”