Perfect Fractional Matchings in Bipartite Graphs Via Proportional Allocations
Daniel Hathcock, R. Ravi
TL;DR
We address when a proportional allocation that uses right-side weights yields a fractional perfect matching in a bipartite graph by linking it to matrix scaling. The core approach shows that an $(S,C)$-scaling of the incidence matrix $A$ exists if and only if the graph is matching-covered, yielding a perfect proportional allocation, and uses the Dulmage–Mendelsohn decomposition to extend to non-matching-covered graphs. The paper also discusses a two-capacity extension, illustrating fundamental limitations with a complete bipartite example and highlighting open questions about simple allocations under multiple constraints. Overall, the work provides a precise combinatorial criterion and a constructive method for perfect proportional allocations in bipartite graphs, with implications for fast online resource allocations and fair distribution of supply to budgets.
Abstract
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.