Envy-freeness and relaxed stability for lower-quotas: A parameterized perspective

We consider the problem of assigning agents to resources under the two-sided pref-erence list setting 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. Krishnaa et al. (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) that satisfies lower quotas is not approximable within a certain constant factor unless P = NP.In this work, we investigate the parameterized complexity of MAXEFM and MAXRSM. We show that MAXEFM is W[1]-hard and MAXRSM is para-NP-hard when parame-terized on several natural parameters derived from the instance. We present kernel-ization results and FPT algorithms for both problems parameterized on other relevant parameters.& COPY; 2023 Elsevier B.V. All rights reserved.