Classified Stable Matching

We introduce the {\sc classified stable matching} problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of ``order types.’’ We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian.

💡 Research Summary

The paper introduces the Classified Stable Matching (CSM) problem, motivated by academic hiring where institutions not only rank applicants but also impose class‑specific lower and upper quotas (e.g., by research area). Formally, each institution (i) partitions the applicant pool into a family of classes (\mathcal{C}i). For every class (c\in\mathcal{C}i) a lower bound (\ell{ic}) and an upper bound (u{ic}) are given, and a matching (M) must satisfy (\ell_{ic}\le |M\cap c|\le u_{ic}) for all (i,c). The goal is to find a stable matching—no applicant‑institution pair or group can block the outcome—while respecting all quota constraints.

The study is divided into two major parts.

1. Order‑type based complexity classification

The authors abstract the inclusion relationships among the classes of each institution as a directed graph, called an order type. Nodes represent classes; an arc (c\to c’) indicates that (c) is a subset of (c’).

If every order type is a downward forest (each node has at most one parent and the graph is acyclic), the paper presents a polynomial‑time algorithm. The algorithm proceeds bottom‑up: it first matches applicants to the most specific (leaf) classes, then aggregates these decisions upward, always respecting the upper bounds that automatically propagate to ancestors and the lower bounds that can be satisfied locally. By carefully ordering proposals according to the Gale‑Shapley framework and using the hierarchical structure to avoid quota violations, the algorithm guarantees a stable matching whenever one exists.

If an order type contains a node with two parents or any cycle (i.e., the class family is not a downward forest), the existence problem becomes NP‑complete. The authors prove this by a reduction from 3‑SAT/Exact‑Cover, encoding variables and clauses as classes and quota constraints so that a satisfying assignment corresponds to a stable matching and vice‑versa. Consequently, the structural property of the class family completely determines tractability.

2. Polyhedral approach for laminar families with only upper bounds

The second part assumes laminar class families (any two classes are either disjoint or one contains the other) and no lower bounds. Under these conditions the authors construct a linear description of the stable matching polytope:

1. Basic matching constraints – each applicant is matched to at most one institution, and each institution‑applicant edge is represented by a binary variable (x_{ij}).
2. Stability constraints – for any unmatched pair ((i,j)) the sum of edges that each side prefers over the other must be at least one, preventing a blocking pair.
3. Class upper‑bound constraints – for each class (c) the sum of variables over edges belonging to (c) does not exceed its quota (u_c).

The authors prove that this system defines an integral polytope: every extreme point is integral. The proof hinges on two observations. First, the matrix formed by the constraints is totally unimodular because laminarity guarantees that each column participates in at most one class‑inequality in a hierarchical manner. Second, any basic feasible solution of the linear program satisfies the matching constraints exactly, forcing all variables to be 0 or 1. This integrality result implies that any linear objective (e.g., maximizing total preference score, minimizing total rank, or any weighted social welfare) can be optimized over the stable matchings in polynomial time using the Ellipsoid method or any LP solver.

A noteworthy corollary is that when all lower bounds are zero and the class families are laminar, the model collapses to the many‑to‑many unclassified stable matching problem. The authors therefore obtain a complete linear description of the many‑to‑many stable matching polytope, answering an open question posed by Sethuraman, Teo, and Qian regarding the exact polyhedral characterization of that setting.

Contributions and implications

1. Complexity dichotomy – The paper identifies a clean boundary: downward‑forest order types admit a polynomial algorithm; any deviation leads to NP‑completeness. This clarifies when class‑specific quotas can be handled efficiently.
2. Integral polyhedral formulation – For laminar families with only upper quotas, the authors give a compact linear system whose feasible region is exactly the set of stable matchings, and they prove its integrality.
3. Algorithmic consequences – The integral description enables the computation of optimal stable matchings for a wide range of objectives via linear programming, extending the classic Gale‑Shapley framework to quota‑constrained, many‑to‑many environments.
4. Resolution of an open problem – By showing that the many‑to‑many stable matching polytope is captured by the same constraints, the paper settles the longstanding question of its linear description.

Future directions

The paper suggests several extensions: handling lower bounds together with upper bounds in laminar families, exploring approximation algorithms for non‑laminar or non‑downward‑forest structures, and incorporating dynamic preferences or stochastic arrivals of applicants. Moreover, empirical evaluation on real hiring data could illuminate the practical performance of the proposed algorithms.

In summary, the work advances both the theoretical understanding of stable matching under class‑specific quotas and provides concrete tools for solving such problems in realistic settings such as academic hiring, medical residency placement, and corporate recruitment.