Bridging the Gap Between Stable Marriage and Stable Roommates: A Parametrized Algorithm for Optimal Stable Matchings

Abstract

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable, one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an {\it optimal} stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding optimal stable marriages in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the \emph{minimum crossing distance}. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find optimal stable matchings efficiently. More generally, we show that for an SR instance with minimum crossing distance $k$, optimal stable matchings can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed parameter tractable (FPT) with respect to minimum crossing distance.

Figures (4)

• Figure 1: Relationship between structure and terminology for SM and SR instances.
• Figure 2: On the left is a mirror poset $\mathcal{P}$ with five dual pairs. On the right is the median graph $G(\mathcal{P})$ formed by its seven complete closed subsets. Two subsets are adjacent in the median graph if and only if they differ by one element.
• Figure 3: At the top left is the median semilattice $\mathcal{L}(\mathcal{P}, W)$ formed by rooting the median graph $G(\mathcal{P})$ in Figure \ref{['figmirror']} at the complete closed subset $W = \{\rho_1, \rho_2, \rho_3, \rho_4, \rho_5\}$. It has three maximal elements. At the bottom left is the mirror poset $\mathcal{P}$ oriented at $W$; the lower half of $\mathcal{P}$ consists of the elements of $W$ and the other half has the elements of $P-W$. Each $\rho_i$ is renamed as $\rho^-_i$ and its dual as $\rho_i^+$. The renamed poset is shown on the bottom right. It has three pairs of crossing edges from $\mathcal{P}^-$ to $\mathcal{P}^+$: $\{(\rho_2^-, \rho_4^+), (\rho_4^-, \rho_2^+)\}$, $\{(\rho_3^-, \rho_4^+), (\rho_4^-, \rho_3^+)\}$ and $\{(\rho_3^-, \rho_5^+), (\rho_5^-, \rho_3^+)\}$. At the top right, the complete closed subsets of $\mathcal{P}$ are updated to reflect the change in the naming convention.
• Figure 4: This time around, the median graph $G(\mathcal{P})$ in Figure \ref{['figmirror']} is rooted at $W = \{\rho_1, \rho_2, \overline{\rho_3}, \rho_4, \rho_5\}$ to create another median semilattice. This semilattice has two maximal elements. By our convention, $\mathcal{P}$ is oriented with $W$ as the base and the rotations are renamed to reflect this orientation. The resulting poset on the right has only one pair of crossing edges: $\{(\rho_2^-, \rho_4^+), (\rho_4^-, \rho_2^+)\}$.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 2.1: Birkhoff1937Rings
• Theorem 2.2: Irving1986ComplexityGusfield1989Stable
• Theorem 2.3: See discussion in Section 3.3 of Gusfield1989Stable
• Theorem 2.4: Irving1987efficient
• Theorem 2.5: Irving1987efficient
• Theorem 3.1: Gusfield1989Stable
• Lemma 3.2
• Theorem 3.3: Gusfield1988StructureIrving1986Stable
• Theorem 3.4: See discussion in Section 4.4.1 in Gusfield1989Stable