Efficient approximation and heuristics for quantile stable matchings

Develop efficient approximation algorithms or heuristics for computing compromise stable matchings within the cycle-based framework for many-to-many markets with strongly substitutable preferences and the law of aggregate demand, despite the computational difficulty of exactly counting ideals in the associated preference-cycle poset.

Background

The paper characterizes quantile stable matchings by associating each stable matching with a closed set of preference cycles and determining the relevant closed set through a counting function on the cycle-precedence poset. Computing exact quantile stable matchings requires counting ideals in this poset, a problem identified as #P-complete. The authors explicitly leave unresolved whether the cycle-based representation can support computationally efficient approximation methods or heuristics for obtaining compromise stable matchings.