# Stable Fractional Matchings

## TL;DR

This paper explores the computational complexity of finding welfare-maximizing stable fractional matchings in a generalized setting with cardinal preferences, revealing hardness results and providing initial approximation algorithms.

## Contribution

It introduces the first complexity results for stable fractional matchings and analyzes approximation algorithms with welfare guarantees.

## Key findings

- Stable fractional matchings can significantly outperform integral ones in social welfare.
- Achieving better approximations than the presented algorithms is computationally hard.
- The paper provides structural insights into stable fractional matchings.

## Abstract

We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). After observing that, in this cardinal setting, stable fractional matchings can have much higher social welfare than stable integral ones, our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) or nearly-optimal stable fractional matching. We present simple approximation algorithms for this problem with weak welfare guarantees and, rather unexpectedly, we furthermore show that achieving better approximations is hard. This computational hardness persists even for approximate stability. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings. En route to these results, we provide a number of structural observations.

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Source: https://tomesphere.com/paper/1902.06698