Minimum Cut Representability of Stable Matching Problems

TL;DR

The paper develops a framework, called Minimum Cut Representability, to solve optimization and feasibility problems over stable matchings by reducing them to minimum $s$-$t$ cuts on rotation-based witness digraphs. It provides a complete characterization based on first- and second-order differentials and sublattice feasibility, and shows how to construct the witness digraph when the conditions hold; the nonlinear objective is captured via a second-order expansion around stable matchings. The framework generalizes the classic Minimum Weight Stable Matching (MWSM) as the linear case (zero second-order differentials) and applies to real-world inspired problems such as Matching Siblings and Two-stage Stochastic Stable Matching, yielding polynomial-time solvability for MSSS and certain MSSP instances, while MSDP is NP-hard and some MSSP instances resist linearization. It also addresses two-stage settings with explicit second-stage distributions in polynomial time, and with implicit distributions via sampling where NP-hardness arises but practical SAA-based approximations are available; empirical results illustrate the method’s advantage over standard one-sided stable matchings and its closeness to hindsight optima. Overall, the work offers a unified, scalable approach for complex stability-constrained optimization in matching markets with applications to education and resource allocation problems.

Abstract

We introduce and study Minimum Cut Representability, a framework to solve optimization and feasibility problems over stable matchings by representing them as minimum s-t cut problems on digraphs over rotations. We provide necessary and sufficient conditions on objective functions and feasibility sets for problems to be minimum cut representable. In particular, we define the concepts of first and second order differentials of a function over stable matchings and show that a problem is minimum cut representable if and only if, roughly speaking, the objective function can be expressed solely using these differentials, and the feasibility set is a sublattice of the stable matching lattice. To demonstrate the practical relevance of our framework, we study a range of real-world applications, including problems involving school choice with siblings and a two-stage stochastic stable matching problem. We show how our framework can be used to help solving these problems.

Figures (6)

• Figure 1: Example adapted from faenza2022legal. Consider the first-stage instance $I_1$ given above, where $a^1,\dots, a^5$ are the students and all schools have a quota of $1$. Then $M_0= \{a^1b^1, a^2b^2, a^3b^3, a^4b^4, a^5b^5\}$ and $\underline M = \{a^1b^4, a^2b^3, a^3b^2, a^4b^1, a^5b^5\}$ are two stable matchings of $I_1$, with $M_0$ being student-optimal (the partner of each student $a$ in $M_0$ and $\underline M$ is, respectively, boxed and underlined in $a$'s preference list). If $b^5$ leaves the market in the second-stage, the only stable matching is $M'= \underline M\setminus \{a^5b^5\}$, hence, $\underline M$ minimizes the number of students downgraded in the second-stage.
• Figure 2: Example of an instance of (MSSP), with $\mathop{\mathrm{\mathcal{C}}}\limits = \{(a_1,\bar{a}_1)\}$ and two activities: $b^{1}_{1}, b^{2}_{1},b^{3}_{1}$ are the classes of activity $c_1$; $b^{1}_{2}, b^{2}_{2}$ are the two classes of activity $c_2$. $a_1$ is eligible for $b^{1}_{1},b^{1}_{2}$; $\bar{a}_1$ is eligible for $b^{2}_{1},b^{2}_{2}$. All classes have a quota of $1$. Missing agents in the preference lists above can be added in any order after the outside option. The activity-stable matchings are the one-sided optimal stable matchings $M_0=\{a_1b^{1}_{1}, \bar{a}_1b^{2}_{1}, a_2b^{3}_{1}, a_3b^{1}_{2},a_4b^{2}_{2}\}$ (both $a_1$ and $\bar{a}_1$ attend a class of activity $c_1$) and $M_z=\{a_1b^{1}_{2}, \bar{a}_1b^{2}_{2}, a_2b^{3}_{1}, a_3b^{1}_{1},a_4b^{2}_{1}\}$ (both $a_1$ and $\bar{a}_1$ attend a class of activity $c_2$), while the other two stable matchings $M_1=\{a_1b^{1}_{2}, \bar{a}_1b^{2}_{1}, a_2b^{3}_{1}, a_3b^{1}_{1},a_4b^{2}_{2}\}$ and $M_2=\{a_1b^{1}_{1}, \bar{a}_1b^{2}_{2}, a_2b^{3}_{1}, a_3b^{1}_{2},a_4b^{2}_{1}\}$ are not activity-stable.
• Figure 3: The digraph $D(I,J_2)$ that our algorithm computes for the deterministic $2$-stage instance with first-stage instance $I$ and second-stage instance $J_2$, defined as in Example \ref{['ex:2-stage']}. We set $\lambda=1$ and $c_1,c_2$ be the $0$ vectors. Nodes on the left correspond to rotations $\rho_1=((a^1b^1,a^2b^3,a^3b^5,a^4b^2),(a^1b^2,a^2b^5,a^3b^1,a^4b^3))$, $\rho_2=((a^4b^3,a^5b^4),(a^4b^4,a^5b^3))$, and $\rho_3=((a^1b^2,a^2b^5,a^3b^1,a^4b^4),(a^1b^4,a^2b^1,a^3b^2,a^4b^5))$ from instance $I$. The node on the right corresponds to the unique rotation $\rho_1'=((a^4b^4,a^5b^5),(a^4b^5,a^5b^4))$ from $J_2$. Arcs of capacity $0$ are not represented. With this special objective function, the objective value of \ref{['eq:exp2sto']} computed in a pair of first- and second-stage stable matchings $M_1$ and $M_2$ is equal to the total rank drop of students between the first- and second-stage. The above digraph is such that the $s-t$ cut corresponding to any pair of first- and second-stage stable matchings $M_1,M_2$ has capacity equal to the objective value of $M_1, M_2$ in \ref{['eq:exp2sto']} plus $2$. For example, the $s$-$t$ cut $\{s,\rho_1,\rho_1'\}$ corresponds to the pair of stable matchings $M_0^{I}/\rho_1=M^{I}_1=\{a^1b^2,a^2b^5,a^3b^1,a^4b^3,a^5b^4\}$, $M^{J_2}_0/\rho_1'=M^{J_2}_1=\{a^1b^1,a^4b^5,a^5b^4\}$ whose objective value in \ref{['eq:exp2sto']} is $2$, and the cut has a value $2+2 = 4$.
• Figure 4: Comparison of the objective value of the optimal stable matching with other stable matchings commonly used in practice, on an instance with $50$ students and $50$ schools with randomly generated preferences.
• Figure 5: An instance of (MSSP) with $\mathop{\mathrm{\mathcal{C}}}\limits =\{(a_i,\bar{a}_i)\}_{i=1,2,3}$ and three activities $c_1, c_2, c_3$, with two classes $b_i,\bar{b}_i$ for each activity $c_i$. Each student $a_i$ (resp., $\bar{a}_i$) is eligible for class $b_1,b_2,b_3$ (resp., $\bar{b}_1,\bar{b}_2,\bar{b}_3$). All classes have quota of $1$.

Theorems & Definitions (63)

• Definition 1: Partially Ordered Set (Poset)
• Definition 2: Meet, Join, Lattice
• Definition 3: Lattice of Upper-Closed Subsets of a Poset
• Definition 4: Lattice Isomorphism
• Definition 5: Sublattice
• Lemma 1: Lattice of Stable Matchings
• Definition 6: Student Optimal and Student Pessimal Stable Matchings
• Definition 7: Rotations
• Lemma 2: Size of ${\cal R}(I)$
• Definition 8: Elimination of a rotation