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. [15] 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.
ACM Subject ClassificationMathematics of computing → Graph theory; Theory of computation → Design and analysis of algorithms