Disjoint Stable Matchings in Linear Time

TL;DR

This paper presents a linear-time algorithm to find the largest collection of pairwise edge-disjoint stable matchings in a given instance, extending classical results and enabling enumeration of maximum-length chains.

Contribution

It introduces a novel linear-time algorithm for disjoint stable matchings and a polynomial-time method to enumerate all maximum-length chains in the stable matching lattice.

Findings

Largest collection of disjoint stable matchings found in linear time

Algorithm to enumerate all maximum-length chains in polynomial time

Derived expected number of chains in random instances

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.