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A poset representation for stable contracts in a two-sided market generated by integer choice functions

Generalizing a variety of earlier problems on stable contracts in two-sided markets, Alkan and Gale introduced in 2003 a general stability model on a bipartite graph $G=(V,E)$ in which the vertices are interpreted as ``agents'', and the edges as possible ``contract'' between pairs of ``agents''. The edges are endowed with nonnegative capacities $b$ giving upper bounds on ``contract intensities'', and the preferencies of each ``agent'' $v\in V$ depend on a \emph{choice function} (CFs) that acts on the set of ``contracts'' involving $v$, obeying three well motivated axioms of \it{consistence}, \it{substitutability} and \it{cardinal monotonicity}. In their model, the capacities and choice functions can take reals or discrete values and, extending well-known earlier results on particular cases, they proved that systems of \it{stable} contracts always exist and, moreover, their set $\cal S$ constitutes a distributive lattice under a natural comparison relation $\prec$. In this paper, we study Alkan--Gale's model when all capacities and choice functions take integer values. We characterize the set of rotations -- augmenting cycles linking neighboring stable assignments in the lattice $(\cal S,\prec)$, explain how to construct the rotations efficiently, and devise a weighted poset in which the lattice of closed functions is isomorphic to $(\cal S,\prec)$, thus obtaining an explicit representation for the latter. We show that in general the size of the poset is at most $b^{\rm max}|E|$, where $b^{\rm max}$ is the maximal capacity, and the poset can be constructed in pseudo polynomial time. Then we explain that by imposing an additional condition on CFs, the size of the poset becomes polynomial in $|V|$, and the total time reduces to a polynomial in $|V|,\log b^{\rm max}$.