Critical Relaxed Stable Matchings with Two-Sided Ties

Abstract

We consider the stable marriage problem in the presence of ties in preferences and critical vertices. The input to our problem is a bipartite graph $G = (\mathcal {A}\cup \mathcal {B}, E)$ where $\mathcal {A}$ and $\mathcal {B}$ denote sets of vertices which need to be matched. Each vertex has a preference ordering over its neighbours possibly containing ties. In addition, a subset of vertices in $\mathcal {A}\cup \mathcal {B}$ are marked as critical and the goal is to output a matching that matches as many critical vertices as possible. Such matchings are called critical matchings in the literature and in our setting, we seek to compute a matching that is critical as well as optimal with respect to the preferences of the vertices.

Stability, which is a well-accepted notion of optimality in the presence of two-sided preferences, is generalized to weak-stability in the presence of ties. It is well known that in the presence of critical vertices, a matching that is critical as well as weakly stable may not exist. Popularity is another well-investigated notion of optimality for the two-sided preference list setting, however, in the presence of ties (even with no critical vertices), a popular matching need not exist. We, therefore, consider the notion of relaxed stability which was introduced and studied by Krishnaa et. al. (SAGT 2020). We show that in our setting a critical matching which is relaxed stable always exists although computing a maximum-sized relaxed stable matching turns out to be NP-hard. Our main contribution is a $\frac{3}{2}$-approximation to the maximum-sized critical relaxed stable matching for the stable marriage problem where ties as well as critical vertices are present on both the sides of the bipartition.