Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that “approximately popular” branchings always exist.

设G是一个有向图，其中每个节点对其传入边都有偏好。节点的偏好自然扩展到对分支的偏好，即有向森林；如果B没有失去反对任何分支的正面选举（节点投票），则分支B很受欢迎。这种流行的分支在流动民主中具有自然的应用。流行分支问题是决定G 是否承认流行分支。我们根据双证书对流行分支进行了表征并使用此表征为流行的分支问题设计一种有效的组合算法。当偏好是弱排名时，我们使用我们的特征来制定原始空间中流行的分支多胞体，并表明我们的算法可以修改以计算具有最小不受欢迎程度的分支。当偏好是严格的排名时，我们表明“近似流行”的分支总是存在的。