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

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