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

We consider the problem of assigning agents to resources under the two-sided preference list model with upper and lower-quota requirements on resources. This 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 satisfies the lower-quotas. Krishnaa et al. [SAGT 2020] explore two alternative optimality notions for the instances with lower-quotas -- envy-freeness and relaxed stability. They investigate the problem of computing a maximum size envy-free matching (MAXEFM) and a maximum size relaxed stable matching (MAXRSM) that satisfies 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 parameterized complexity of MAXEFM and MAXRSM. We consider several natural parameters derived from the instance -- the number of resources with non-zero lower-quota, deficiency of the instance, maximum length of the preference list of a resource with non-zero lower-quota, among others. We show that MAXEFM problem is W[1]-hard for several interesting parameters and MAXRSM problem is para-NP-hard for two natural parameters. We present a kernelization result on a combination of parameters and FPT algorithms for both problems.