Abstract
Suppose that each member of a set of agents has a preference list of a subset of houses, possibly involving ties, and each agent and house has their capacity denoting the maximum number of houses/agents (respectively) that can be matched to him/her/it. We want to find a matching M, called popular, for which there is no other matching M′ such that more agents prefer M′ to M than M to M′, subject to a suitable definition of ”prefers”. In the above problem each agent uses exactly one vote to compare two matchings. In the other problem we consider in the paper each agent has a number of votes equal to their capacity. Given two matchings M and M′, an agent compares their best house in matching M ∖ (M ∩ M′) to their best house in matching M′ ∖ (M ∩ M′) and gives one vote accordingly, then their second best houses and so on. A matching M for which there is no matching M′ such that M′ gets a bigger number of votes than M, when M and M′ are compared in the way described above, is then called clan-popular. Popular matchings have been studied quite extensively, especially in the one-to-one setting. In the many-to-many setting we provide a characterisation of popular and clan-popular matchings, show NP-hardness results for very restricted cases of the above problems and for certain versions describe novel polynomial algorithms. The given characterisation is also valid for popular matchings in the one-to-one setting.
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Paluch, K. (2012). Popular and Clan-Popular b-Matchings. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_15
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