Finding popular branchings in vertex-weighted directed graphs.

Popular matchings have been intensively studied recently as a relaxed concept of stable matchings. By applying the concept of popular matchings to branchings in directed graphs, Kavitha et al. introduced popular branchings. Let G = (V G , E G) be a directed graph, where each vertex has preferences over its incoming arcs, and B and B ′ be branchings in G. A vertex v ∈ V G prefers B to B ′ if v prefers its incoming arc of B to that of B ′ , where having an arbitrary incoming arc is preferred to having no incoming arc. We say that B is more popular than B ′ if the number of vertices preferring B to B ′ is greater than the number of vertices preferring B to B ′. A branching B is called a popular branching if there is no branching more popular than B. Kavitha et al. proposed an algorithm for finding a popular branching when the preferences of each vertex are given by a strict partial order. The correctness of this algorithm is proved by utilizing classical theorems on the duality of weighted arborescences. In this paper, we generalize popular branchings to weighted popular branchings in vertex-weighted directed graphs, in the same manner as weighted popular matchings by Mestre. We give an algorithm for finding a weighted popular branching for the case where the preferences of each vertex are given by a total preorder and the weights satisfy certain conditions. Our algorithm is based on Kavitha et al.'s algorithm, while it includes elaborated procedures resulting from the vertex-weights. Its correctness also builds upon the duality of weighted arborescences. [ABSTRACT FROM AUTHOR]