Ex-post Stability under Two-Sided Matching: Complexity and Characterization

TL;DR

This paper investigates the computational complexity of ex-post stability in two-sided matching problems, revealing NP-completeness with ties and dichotomous preferences, while providing polynomial algorithms for stronger stability notions.

Contribution

It characterizes the complexity of testing ex-post stability under various preference structures and offers algorithms for certain stability variants.

Findings

Testing ex-post stability is NP-complete with ties or dichotomous preferences.

An integer programming algorithm can find decompositions with maximum weakly stable probability.

Stronger stability notions like robust ex-post stability can be tested in polynomial time.

Abstract

A probabilistic approach to the stable matching problem has been identified as an important research area with several important open problems. When considering random matchings, ex-post stability is a fundamental stability concept. A prominent open problem is characterizing ex-post stability and establishing its computational complexity. We investigate the computational complexity of testing ex-post stability. Our central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete. The result even holds if both sides have dichotomous preferences. On the positive side, we give an algorithm using an integer programming approach, that can determine a decomposition with a maximum probability of being weakly stable. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.