Perfect Fractional Matchings in Bipartite Graphs Via Proportional Allocations

Given a bipartite graph that has a perfect matching, a prefect proportional allocation is an assignment of positive weights to the nodes of the right partition so that every left node is fractionally assigned to its neighbors in proportion to their weights, and these assignments define a fractional perfect matching. We prove that a bipartite graph has a perfect proportional allocation if and only if it is matching covered, by using a classical result on matrix scaling. We also present an extension of this result to provide simple proportional allocations in non-matching-covered bipartite graphs.