[Paper Review] Popular Matching with Lower Quotas

Meghana Nasre, Prajakta Nimbhorkar|arXiv (Cornell University)|Apr 25, 2017

Game Theory and Voting Systems3 references6 citations

TL;DR

This paper introduces a novel algorithmic framework for the Hospital-Resident problem with Lower Quotas (HRLQ), proving that a feasible matching popular among all feasible matchings always exists—even when no stable matching is possible. The approach reduces HRLQ instances to standard Hospital-Resident problems via tailored transformations, enabling efficient computation of maximum cardinality popular matchings in O(|R|(|E| + |H|)) time and popular matchings among maximum cardinality feasible matchings in O(|R|²(|E| + |H|)) time.

ABSTRACT

We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph $G = (\mathcal{R} \cup \mathcal{H}, E)$ where $\mathcal{R}$ and $\mathcal{H}$ denote sets of residents and hospitals respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital $h$ has an associated upper-quota $q^+(h)$ and lower-quota $q^-(h)$. A matching $M$ in $G$ is an assignment of residents to hospitals, and $M$ is said to be feasible if every resident is assigned to at most one hospital and a hospital $h$ is assigned at least $q^-(h)$ and at most $q^+(h)$ residents. Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching $M$ is popular if no other matching $M'$ gets more votes than $M$ when vertices vote between $M$ and $M'$. When there are no lower quotas, there always exists a stable matching and it is known that every stable matching is popular. We show that in an HRLQ instance, although a feasible stable matching need not exist, there is always a matching that is popular in the set of feasible matchings. We give an efficient algorithm to compute a maximum cardinality matching that is popular amongst all the feasible matchings in an HRLQ instance.

Motivation & Objective

• Address the practical challenge of assigning residents to hospitals when lower quotas constrain feasible matchings, especially when no stable matching exists.
• Relax the stability requirement with popularity—a stronger, more robust notion of fairness and collective satisfaction.
• Ensure that the output matching is both feasible (respects lower and upper quotas) and popular among all feasible matchings.
• Develop efficient algorithms to compute maximum cardinality popular matchings and popular matchings within maximum cardinality feasible matchings in HRLQ.
• Provide a combinatorial, non-LP-based proof of popularity, generalizing and extending prior results in stable matching and HR literature.

Proposed method

• Construct a transformed HR instance G′ by modifying the preference lists and capacities of hospitals to encode lower quota constraints.
• Use a level-based decomposition of residents and hospitals to model matching levels and identify critical path structures in alternating paths and cycles.
• Define voting rules where residents and hospitals compare matchings via preference-based votes: (1,1) for mutual preference, (−1,−1) for mutual disfavor.
• Apply the Gale-Shapley algorithm to the transformed instance G′ to compute a stable matching, which is then mapped back to a feasible matching in the original HRLQ instance.
• For maximum cardinality popular matchings, use a reduction to a second HR instance G′′ with modified preference priorities to favor unmatched residents.
• Prove correctness via structural lemmas showing that mapped matchings preserve feasibility and popularity through path decomposition and vote balance arguments.

Experimental results

Research questions

• RQ1 Does a feasible matching that is popular among all feasible matchings always exist in an HRLQ instance, even when no stable matching exists?
• RQ2 Can a maximum cardinality popular matching in the set of feasible matchings be computed efficiently in HRLQ?
• RQ3 Can a popular matching be computed among all maximum cardinality feasible matchings in HRLQ, and what is the computational complexity?
• RQ4 How can the notion of popularity be adapted to HRLQ while preserving combinatorial structure and avoiding reliance on linear programming?
• RQ5 What structural properties of alternating paths and cycles in M ⊕ N ensure that a matching M is popular?

Key findings

• A feasible matching that is popular among all feasible matchings always exists in any HRLQ instance, even when no stable matching exists.
• An O(|R|(|E| + |H|)) time algorithm computes a maximum cardinality matching that is popular among all feasible matchings.
• An O(|R|²(|E| + |H|)) time algorithm computes a matching that is popular among all maximum cardinality feasible matchings.
• The algorithm constructs a stable matching in a transformed HR instance G′, which is then mapped back to a feasible and popular matching in the original HRLQ instance.
• The proof of popularity relies on a vote-balance argument over alternating paths and cycles, showing that (1,1) edges are at most (−1,−1) edges plus a level-dependent offset.
• The structural lemmas (e.g., Lemma 13) ensure that hospitals and residents are correctly partitioned across levels, and that no invalid edges exist across levels, preserving feasibility and popularity.