Envy-freeness and Relaxed Stability for Lower-Quotas : A Parameterized Perspective

TL;DR

This paper studies the computational complexity of finding large envy-free and relaxed stable matchings in resource allocation problems with quotas, providing hardness results and fixed-parameter tractable algorithms.

Contribution

It is the first to analyze the parameterized complexity of MAXEFM and MAXRSM, offering kernelization and FPT algorithms for various parameters.

Findings

MAXEFM is W[1]-hard under certain parameters.

MAXRSM is para-NP-hard under certain parameters.

Kernelization and FPT algorithms are developed for other parameters.

Abstract

We consider the problem of assigning agents to resources under the two-sided preference list model where resources specify an upper-quota and a lower-quota, that is, respectively the maximum and minimum number of agents that can be assigned to it. Different notions of optimality including envy-freeness and relaxed stability are investigated for this setting and the goal is to compute a largest optimal matching. Krishnaa et al. [SAGT 2020] show that in this setting, the problem of computing a maximum size envy-free matching (MAXEFM) or a maximum size relaxed stable matching (MAXRSM) is not approximable within a certain constant factor unless P = NP. This work is the first investigation of parameterized complexity of MAXEFM and MAXRSM. We show that MAXEFM is W [1]-hard and MAXRSM is para-NP-hard when parameterized on several natural parameters derived from the instance. We present kernelization results and FPT algorithms for both problems parameterized on other relevant parameters.