The Strongly Stable Matching Problem with Closures

TL;DR

This work studies the strongly stable matching problem with closures in a bipartite setting, where a subset $S$ of hospitals closes when unmatched. It proves NP-completeness of deciding existence of a stable matching under restricted conditions, and offers two polynomial-time solvable regimes: separated preferences and bounded-degree instances. A central preprocessing lemma yields a core edge set $R$ and a matching $\\mu$ that enable a decomposition of the residual instance into tractable subproblems, including a path-based construction. The results connect strongly stable matchings with envy-free matchings and delineate tractable regions, advancing the understanding of stability with closures and providing practical algorithms for important special cases.

Abstract

In this paper, we consider one-to-one matchings between two disjoint groups of agents. Each agent has a preference over a subset of the agents in the other group, and these preferences may contain ties. Strong stability is one of the stability concepts in this setting. In this paper, we consider the following variant of strong stability, and we consider computational complexity issues for this solution concept. In our setting, we are given a subset of the agents on one side. Then when an agent in this subset is not matched to any partner, this agent cannot become a part of a blocking pair. In this paper, we first prove that the problem of determining the existence of a stable matching in this setting is NP-complete. Then we give two polynomial-time solvable cases of our problem. Interestingly, one of our positive results gives a unified approach to the strongly stable matching problem and the envy-free matching problem.

Figures (5)

• Figure 1: (a) The gadget for the variable $\alpha_i$. (b) The gadget for the clause $C_t$. (c) $\ell_{1,1} = \neg \alpha_1$ and $\ell_{1,2} = \alpha_2$.
• Figure 2: (a) $j_t = 1$. (b) $j_t = 2$. (c) $j_t = 3$.
• Figure 3: (a) $\phi(\alpha_1) = 1$, $\phi(\alpha_2) = 1$, and $j_1 = 2$. (b) $\phi(\alpha_1) = 0$, $\phi(\alpha_2) = 1$, and $j_1 = 1$.
• Figure 4: (a) An example of $G_X$. (b) The tree rooted at $h_4$. (c) The desired matching.
• Figure 5: (a) The real lines represent edges in $L$. The broken lines represent edges in $E \setminus L$. The bold lines represent edges in $\mu$. (b) The bold lines represent edges in $\sigma$.

Theorems & Definitions (49)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5