Critical Relaxed Stable Matchings with Ties in the Many-to-Many Setting

TL;DR

This work addresses maximum-size CRITICAL-RSM in many-to-many bipartite matchings with ties and lower quotas on both sides. It combines a generalized Király-style proposal algorithm for ties with a multi-level relaxed stability framework, certified through a cloned graph construction $G_M$ to prove correctness. The authors prove existence of CRITICAL-RSM, establish NP-hardness for the optimization problem, and present a polynomial-time $\frac{3}{2}$-approximation achieving at least $\frac{2}{3}$ of the optimum. The results extend relaxed-stability notions to the complex setting of bi-directional quotas and ties, enabling quota-respecting, preference-based assignments in practical applications such as education and workforce allocation.

Abstract

We study the many-to-many bipartite matching problem in the presence of preferences where ties, as well as lower quotas, may appear on both sides of the bipartition. The input is a bipartite graph $G=(A \cup B, E)$, where each vertex in $A \cup B$ has a positive upper quota and a non-negative lower quota denoting the maximum and minimum number of vertices that can be assigned to it from its neighborhood. Additionally, each vertex specifies a preference ordering, possibly containing ties, over its neighbors. A \textit{critical} matching is a matching which fulfills vertex lower quotas to the maximum possible extent. We seek to compute a matching that is critical as well as optimal with respect to the preferences of vertices. Stability, a well-accepted notion of optimality in the presence of two-sided preferences, is generalized to weak-stability in the presence of ties. However, 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 model; however, in the presence of ties (even without lower quotas), a popular matching may not exist. We, therefore, consider the notion of relaxed stability, which was introduced and studied by Krishnaa, Limaye, Nasre, and Nimbhorkar~(JoCO 2023). We show that a critical matching that is relaxed stable always exists, although computing a maximum-size relaxed stable matching turns out to be NP-hard. Our main contribution is a $\frac{3}{2}$-approximation algorithm for computing a maximum-size critical relaxed stable matching.

Figures (2)

• Figure 1: The graph $G_M$ after re-arranging the vertices based on their levels in $M$. Red nodes represent critical clones, and black nodes represent non-critical clones. Matched clones in $M^*$ are represented by solid nodes, and unmatched vertices are represented by hollow nodes. The blue horizontal lines represent matched edges in $M^*$. Solid black lines represent edges which are not matched in $M^*$. Note that no edge in $G_M$ is of the form $\mathcal{A}_x\times \mathcal{A}_y$ for $y\le x-2$.
• Figure 2: Blue colored edges denote the edges in $M^*$ whereas red edges denote the edges in $N^*$.

Theorems & Definitions (21)

• Definition 1: Critical Matching
• Definition 2: Stable Matchings
• Definition 3: Relaxed Stability in HRLQ krishnaa2023envy
• Theorem 1
• Definition 4: Uncertain Proposal
• Definition 5: Favorite Neighbor of $a$
• Claim 1
• Definition 6: Relaxed Stability in the Many-to-Many Setting
• Claim 2
• Definition 7: Favorite Neighbor of $a^{\ell}$ for $\ell\in\{{t},{t}^*\}$