Disjoint Stable Matchings in Linear Time

Abstract: We show that given a SM instance G as input we can find a largest collection of pairwise edge-disjoint stable matchings of G in time linear in the input size. This extends two classical results: 1. The Gale-Shapley algorithm, which can find at most two ("extreme") pairwise edge-disjoint stable matchings of G in linear time, and 2. The polynomial-time algorithm for finding a largest collection of pairwise edge-disjoint perfect matchings (without the stability requirement) in a bipartite graph, obtained by combining K\"{o}nig's characterization with Tutte's f-factor algorithm. Moreover, we also give an algorithm to enumerate all maximum-length chains of disjoint stable matchings in the lattice of stable matchings of a given instance. This algorithm takes time polynomial in the input size for enumerating each chain. We also derive the expected number of such chains in a random instance of Stable Matching.