FPT-Approximability of Stable Matching Problems

TL;DR

This paper investigates the parameterized approximability of three stable matching problems, establishing hardness results and providing an FPT-approximation scheme for one problem involving ties.

Contribution

It proves W[1]-hardness for approximating two problems related to blocking pairs and offers an FPT-approximation scheme for a stable matching problem with ties.

Findings

W[1]-hard to approximate to any function of eta for two problems

No FPT-approximation scheme for certain problems unless FPT=W[1]

Provides an FPT-approximation scheme for Max-SMTI with ties

Abstract

We study parameterized approximability of three optimization problems related to stable matching: (1) Min-BP-SMI: Given a stable marriage instance and a number k, find a size-at-least-k matching that minimizes the number $\beta$ of blocking pairs; (2) Min-BP-SRI: Given a stable roommates instance, find a matching that minimizes the number $\beta$ of blocking pairs; (3) Max-SMTI: Given a stable marriage instance with preferences containing ties, find a maximum-size stable matching. The first two problems are known to be NP-hard to approximate to any constant factor and W[1]-hard with respect to $\beta$, making the existence of an EPTAS or FPT-algorithms unlikely. We show that they are W[1]-hard with respect to $\beta$ to approximate to any function of $\beta$. This means that unless FPT=W[1], there is no FPT-approximation scheme for the parameter $\beta$. The last problem (Max-SMTI) is known to be NP-hard to approximate to factor-29/33 and W[1]-hard with respect to the number of ties. We complement this and present an FPT-approximation scheme for the parameter "number of agents with ties".