On stable assignments generated by choice functions of mixed type

Abstract: We consider one variant of stable assignment problems in a bipartite graph endowed with nonnegative capacities on the edges and quotas on the vertices. It can be viewed as a generalization of the stable allocation problem introduced by Ba\H{\i}ou and Balinsky, which arises when strong linear orders of preferences on the vertices in the latter are replaced by weak ones. At the same time, our stability problem can be stated in the framework of a theory by Alkan and Gale on stable schedule matchings generated by choice functions of a wide scope. In our case, the choice functions are of a special, so-called mixed, type. The main content of this paper is devoted to a study of rotations in our mixed model, functions on the edges determining elementary'' transformations between close stable assignments. These look more sophisticated compared with rotations in the stable allocation problem (which are generated by simple cycles). We efficiently construct a poset of rotations and show that the stable assignments are in bijection with the so-called closed functions for this poset; this gives rise to acompact'' affine representation for the lattice of stable assignments and leads to an efficient method to find a stable assignment of minimum cost.