We consider the problem of assigning agents to programs in the presence of two-sided preferences . Each agent has to be assigned to at most one program and each program can accommodate multiple agents. However unlike the standard setting of school choice, we do not have fixed upper quotas with programs as an input -- instead we abstract it as a cost associated with every program . This setting enables the programs to control the number of agents assigned to them and is also applicable in the two-round setting (Gajulapalli et al. , FSTTCS 2020) where largest stable extension is computed. In this setting, our goal is to compute a min-cost stable matching that matches all the agents . We show that the problem is NP-hard even under very severe restrictions on the instance. We complement our negative results by presenting approximation algorithms for general case and a fast exponential algorithm for a special case .
We consider the problem of assigning agents to programs in the presence of two-sided preferences , commonly known as the Hospital Residents problem. In the standard setting each program has a rigid upper quota which cannot be violated. Motivated by applications where quotas may be flexible, we propose and study the problem of computing stable matchings under flexible quotas -- denoted as the SMFQ setting. In the SMFQ setting we have a cost associated with every program and these costs control the quotas. Our goal is to compute a stable matching that matches all agents and is optimal with respect to the cost criteria. We study two optimization problems with respect to the costs and show that there is a sharp contrast in the complexity status of the two optimization criteria. We also establish the connection of our model with the stable extension problem in an apparently different two-round setting of the stable matching problem. We show that our results in the SMFQ setting generalize the stable extension problem .
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. Each agent has to be assigned to at most one program and each program can accommodate multiple agents. However unlike
, commonly known as the Hospital Residents problem. In
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of school choice, we do not have fixed upper quotas with programs as an input
each program has a rigid upper quota which cannot be violated. Motivated by applications where quotas may be flexible, we propose and study the problem of computing stable matchings under flexible quotas
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instead we abstract it as
denoted as the SMFQ setting. In the SMFQ setting we have
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. This setting enables the programs to control the number of agents assigned to them and is also applicable in the two-round setting (Gajulapalli et al. , FSTTCS 2020) where largest stable extension is computed. In this setting, our
and these costs control the quotas. Our
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min-cost
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the agents . We show that the problem is NP-hard even under very severe restrictions on the instance. We complement our negative results by presenting approximation algorithms for general case and a fast exponential algorithm for a special case
agents and is optimal with respect to the cost criteria. We study two optimization problems with respect to the costs and show that there is a sharp contrast in the complexity status of the two optimization criteria. We also establish the connection of our model with the stable extension problem in an apparently different two-round setting of the stable matching problem. We show that our results in the SMFQ setting generalize the stable extension problem