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

We consider the problem of assigning agents to resources in the two-sided preference model with upper and lower-quota requirements on resources. This setting (known as the HRLQ setting) models real-world applications like assigning students to colleges or courses, resident doctors to hospitals and so on. In presence of lower-quotas, an instance may not admit a stable matching that fulfils the lower-quotas. Prem Krishnaa et al. [SAGT 2020] study two alternative notions of optimality for the HRLQ instances -- envy-freeness and relaxed stability. They investigate the complexity of computing a maximum size envy-free matching (MAXEFM) and a maximum size relaxed stable matching(MAXRSM) that fulfils the lower-quotas. They show that both these optimization problems are NP-hard and not approximable within a constant factor unless P=NP. In this work, we investigate the parameterized complexity of MAXEFM and MAXRSM. We consider natural parameters derived from the instance -- the number of lower-quota hospitals, deficiency of the instance, size of a maximum matching, size of a stable matching, length of the preference list of a lower-quota hospital, to name a few. We show that MAXEFM problem is W[1]-hard for several interesting parameters but admits a polynomial size kernel for a combination of parameters. We show that MAXRSM problem does not admit an FPT algorithm unless P=NP for two natural parameters but admits a polynomial size kernel for a combination of parameters in a special case. We also show that both these problems admit FPT algorithms on a set of parameters.