A Faster Algorithm for the Strongly Stable b-Matching Problem

Abstract

We study a generalisation of the stable matching problem to the many-to-many variant in which vertices of the bipartite graph $G = (A \cup B, E)$ may involve ties in their preference lists. We investigate the notion of strong stability and give an $O(m\sum _{y \in B}b(y))$ algorithm for computing a strongly stable b-matching optimal for vertices of A, where we denote $m = |E|$. Our result improves on the previous algorithm by Chen and Ghosh [2]. The main technique allowing us to speed up the algorithm is a generalisation of the notion of level-maximal matchings [8] to the case of b-matchings.

As a byproduct of our results we obtain an O(nm) algorithm for a many-to-one restriction of the problem also known as the hospitals-residents problem, where we denote $n = |A \cup B|$, $m = |E|$. The previous best algorithm had an $O(m\sum _{y \in B} b(y))$ runtime [8].

Partly supported by Polish National Science Center grant 2018/29/B/ST6/02633.