Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph G=(V,E) with strict preferences on the neighbors of the nodes, a matching M is popular if there is no other matching M' such that the number of nodes preferring M' is more than those preferring M. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time O(|E|). Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which are a special class of popular matchings.

翻译：流行性是一种机制设计方法，基于节点的投票在图中寻找公平结构。流行匹配是稳定匹配的松弛：给定一个图 G=(V,E)，节点对其邻居具有严格偏好顺序，若不存在另一个匹配 M' 使得偏好 M' 的节点数量多于偏好 M 的节点数量，则匹配 M 是流行的。本文考虑流行性检验问题，即判断给定匹配是否流行。以往算法通过归约到最大权匹配求解。我们提出一种新算法，将检验流行性的问题归约到最大匹配检验问题，从而实现了 O(|E|) 的线性运行时间。基于线性规划的流行性刻画常用于证明特定匹配的流行性。作为算法的一个结果，我们得到比以往更具结构化的对偶证据。基于这一结果，我们给出了分数流行匹配的组合刻画——分数流行匹配是流行匹配的一个特殊类别。