A Natural Generalization of Stable Matching Solved via New Insights into Ideal Cuts

We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give the first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results:

• We show that the set, ${\mathcal{M}}^{\mathcal{\prime }}$, of maximum weight stable matchings forms a sublattice ${\mathcal{L}}^{\mathcal{\prime }}$ of the lattice $\mathcal{L}$ of all stable matchings.
• We give an efficient algorithm for finding boy-optimal and girl-optimal matchings in ${\mathcal{M}}^{\mathcal{\prime }}$.
• We generalize the notion of rotation, a central structural notion in the context of the stable matching problem, to meta-rotation. Just as rotations help traverse the lattice of all stable matchings, macro-rotations help traverse the sublattice over ${\mathcal{M}}^{\mathcal{\prime }}$.