Integral Stable Allocation Problem on Graphs

As a generalisation of the stable matching problem Bäıou and Balinski [3] defined the stable allocation problem for bipartite graphs, where both the edges and the vertices may have capacities. They constructed a so-called inductive algorithm, that always finds a stable allocation in strongly polynomial time. Here, we generalise their algorithm for non-bipartite graphs with integral capacities. We show that the algorithm does not remain polynomial, although we also present a scaling technique that makes the algorithm weakly polynomial.