Weighted Popular Matchings

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′...