Popular b-matchings

In the paper we study popular b-matchings, which in other words are popular many-to-many matchings. The problem can be best described in graph terms: We are given a bipartite graph G = (A ∪H,E), a capacity function on vertices b : A ∪H → N and a rank function on edges r : E → N . A stands for the set of agents and H for the set of houses. Each member of a set of agents a ∈ A has a preference list Pa of a subset Ha of houses H . For a ∈ A and h ∈ H edge e = (a, h) belongs to E iff h is on Pa and r((a, h)) = i reads that h belongs to (one of) a’s ith choices. We say that a prefers h1 to h2 (or ranks h1 higher than h2) if r((a, h1)) < r((a, h2)). If r(e1) < r(e2) we say that e1 has a higher rank than e2. If there exist a ∈ A and h1, h2 ∈ Ha, h1 6= h2 such that r(e1 = (a, h1)) = r(e2 = (a, h2)), then we say that e1, e2 belong to a tie and graph G contains ties. Otherwise we say that G does not contain ties. A b-matching M of G is such a subset of edges that degM (v) ≤ b(v) for every v ∈ A∪H , meaning that every vertex v has at most b(v) edges of M incident with it. Let r denote the greatest rank (i.e. the largest number) given to any edge of E. With each agent a and each b-matching M we associate a signature denoted as sigM (a), which is an r-tuple (x1, x2, . . . , xr) such that xi (1 ≤ i ≤ r) is equal to the number of edges of rank i matched to a in a b-matching M . We introduce a lexicographic order ≻ on signatures as follows. We will say that (x1, x2, . . . , xr) ≻ (y1, y2, . . . , yr) if there exists j such that 1 ≤ j ≤ r and for each 1 ≤ i ≤ j − 1 there is xi = yi and xj > yj . We say that an agent a prefers b-matching M ′ to M if sigM ′(a) ≻ sigM (a). M ′ is more popular than M , denoted by M ′ ≻ M , if the number of agents that prefer M ′ to M exceeds the number of agents that prefer M to M ′.