[Paper Review] Socially stable matchings in the Hospitals / Residents problem

Georgios Askalidis, Nicole Immorlica+3 more|arXiv (Cornell University)|Mar 8, 2013

Game Theory and Voting Systems26 references4 citations

TL;DR

This paper introduces the Hospitals/Residents problem under Social Stability (HRSS), where blocking pairs are restricted by social network ties—only pairs with existing social connections can block a matching. It proves that finding a maximum socially stable matching is NP-hard but approximable within 3/2, and provides fixed-parameter tractable algorithms for instances with few acquainted pairs.

ABSTRACT

In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings. In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem.

Motivation & Objective

• To address the limitation of traditional stable matching in real-world settings where blocking pairs require social ties to materialize.
• To formalize a new stability concept—social stability—that incorporates social network structures into the Hospitals/Residents problem.
• To investigate the computational complexity of finding maximum socially stable matchings under this new model.
• To develop efficient algorithms for special cases, particularly when the number of acquainted resident-hospital pairs is small.
• To explore structural and algorithmic properties of socially stable matchings, including approximability and fixed-parameter tractability.

Proposed method

• Model the HRSS problem by augmenting the standard HR instance with a social network graph, where edges represent existing social ties between residents and hospitals.
• Define a social blocking pair as a resident-hospital pair that prefers each other over their current matches and are connected in the social network graph.
• Introduce the concept of social stability: a matching is socially stable if no such social blocking pair exists.
• Prove that the problem of finding a maximum socially stable matching is NP-hard via a reduction from a known NP-hard problem.
• Propose a 3/2-approximation algorithm based on maximum weight matching and cardinality augmentation in a transformed graph.
• Design a fixed-parameter tractable (FPT) algorithm for instances where the number of acquainted pairs is bounded by a constant, using a path-based search over a decision tree with O(2^k) leaf nodes.

Experimental results

Research questions

• RQ1 Can a relaxed stability notion that accounts for social ties between agents yield larger matchings than traditional stability?
• RQ2 What is the computational complexity of finding a maximum socially stable matching in the HRSS model?
• RQ3 Is the problem of finding a maximum socially stable matching approximable within a constant factor?
• RQ4 Can fixed-parameter tractable algorithms be developed for HRSS when the number of acquainted pairs is small?
• RQ5 How does the structure of socially stable matchings compare to that of standard stable matchings in HR?

Key findings

• The problem of finding a maximum socially stable matching in HRSS is NP-hard, even when the social network is sparse.
• A 3/2-approximation algorithm exists for the maximum socially stable matching problem, providing a constant-factor guarantee.
• When the number of acquainted pairs is bounded by a constant k, a fixed-parameter tractable algorithm computes a maximum socially stable matching in O(c_max * m * sqrt(n_1 + C)) time.
• The algorithm runs in polynomial time for instances with a constant number of acquainted pairs, making it efficient for small-scale or structured HRSS instances.
• The problem remains NP-hard even when only the number of unacquainted pairs is bounded, indicating inherent complexity in the general case.
• The set of socially stable matchings may vary in size, and no structural characterization similar to the lattice of stable matchings in classical HR is known to exist.