Problem Definition
Consider the problem of matching a set of individuals X to a set of items Y where each individual has a weight and a personal preference over the items. The objective is to construct a matching M that is stable in the sense that there is no matching M′ such that the weighted majority vote will choose M′ over M.
More formally, a bipartite graph \( { (X,Y,E) } \), a weight \( { w(x) \in R^+ } \) for each individual \( { x \in X } \), and a rank function \( { r: E \rightarrow \{1, \dots, |Y|\} } \) encoding the individual preferences are given. For every applicant x and items \( { y_1, y_2 \in Y } \) say applicant x prefers y 1 over y 2 if \( { r(x,y_1) < r(x,y_2) } \), and x is indifferent between y 1 and y 2 if \( { r(x,y_1) = r(x,y_2) } \). The preference lists are said to be strictly ordered if applicants are never indifferent between two items, otherwise the preference lists are said to contain ties.
Let M and M′ be two matchings. An applicant x prefers M over M′...
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Mestre, J. (2008). Weighted Popular Matchings. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_477
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