
Envyfreeness and Relaxed Stability for LowerQuotas: A Parameterized Perspective
We consider the problem of assigning agents to resources under the twos...
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How good are Popular Matchings?
In this paper, we consider the Hospital Residents problem (HR) and the H...
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Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties
Motivated by a serious issue that hospitals in rural areas suffer from s...
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Stable Matchings with Restricted Preferences: Structure and Complexity
It is well known that every stable matching instance I has a rotation po...
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Learning to Order Things
There are many applications in which it is desirable to order rather tha...
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Parameterized inapproximability of Morse matching
We study the problem of minimizing the number of critical simplices from...
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A FineGrained View on Stable ManyToOne Matching Problems with Lower and Upper Quotas
In the Hospital Residents problem with lower and upper quotas (HRQ^U_L)...
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MaximumSize Envyfree Matchings
We consider the problem of assigning residents to hospitals when hospitals have upper and lower quotas. Apart from this, both residents and hospitals have a preference list which is a strict ordering on a subset of the other side. Stability is a wellknown notion of optimality in this setting. Every HospitalResidents (HR) instance without lower quotas admits at least one stable matching. When hospitals have lower quotas (HRLQ), there exist instances for which no matching that is simultaneously stable and feasible exists. We investigate envyfreeness which is a relaxation of stability for such instances. Yokoi (ISAAC 2017) gave a characterization for HRLQ instances that admit a feasible and envyfree matching. Yokoi's algorithm gives a minimum size feasible envyfree matching, if there exists one. We investigate the complexity of computing a maximum size envyfree matching in an HRLQ instance (MAXEFM problem) which is equivalent to computing an envyfree matching with minimum number of unmatched residents (MINUREFM problem). We show that both the MAXEFM and MINUREFM problems for an HRLQ instance with arbitrary incomplete preference lists are NPhard. We show that MAXEFM cannot be approximated within a factor of 21/19. On the other hand we show that the MINUREFM problem cannot be approximated for any alpha > 0. We present 1/(L+1) approximation algorithm for MAXEFM when quotas are at most 1 where L is the length of longest preference list of a resident. We also show that both the problems become tractable with additional restrictions on preference lists and quotas. We also investigate the parameterized complexity of these problems and prove that they are W[1]hard when deficiency is the parameter. On the positive side, we show that the problems are fixed parameter tractable for several interesting parameters.
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