Our input is a directed, rooted graph $G = (V \cup \{r\},E)$ where each vertex in $V$ has a partial order preference over its incoming edges. The preferences of a vertex extend naturally to preferences over arborescences rooted at $r$. We seek a popular arborescence in $G$, i.e., one for which there is no "more popular" arborescence. Popular arborescences have applications in liquid democracy or collective decision making; however, they need not exist in every input instance. The popular arborescence problem is to decide if a given input instance admits a popular arborescence or not. We show a polynomial-time algorithm for this problem, whose computational complexity was not known previously. Our algorithm is combinatorial, and can be regarded as a primal-dual algorithm. It searches for an arborescence along with its dual certificate, a chain of subsets of $E$, witnessing its popularity. In fact, our algorithm solves the more general popular common base problem in the intersection of two matroids, where one matroid is the partition matroid defined by any partition $E = \bigcup_{v\in V} \delta(v)$ and the other is an arbitrary matroid on $E$ of rank $|V|$, with each $v \in V$ having a partial order over elements in $\delta(v)$. We extend our algorithm to the case with forced or forbidden edges. We also study the related popular colorful forest (or more generally, the popular common independent set) problem where edges are partitioned into color classes, and the task is to find a colorful forest that is popular within the set of all colorful forests. For the case with weak rankings, we formulate the popular colorful forest polytope, and thus show that a minimum-cost popular colorful forest can be computed efficiently. By contrast, we prove that it is NP-hard to compute a minimum-cost popular arborescence, even when rankings are strict.

翻译：我们考虑一个有向带根图 $G = (V \cup \{r\},E)$，其中每个顶点 $v \in V$ 对其入边具有偏序偏好。顶点的偏好自然地扩展为对以 $r$ 为根的有向树的偏好。我们寻找 $G$ 中的流行有向树，即不存在"更流行"的有向树。流行有向树在流动民主或集体决策中具有应用；然而，它们并非在所有输入实例中都存在。流行有向树问题旨在判定给定输入实例是否存在流行有向树。我们给出了该问题的一个多项式时间算法，其计算复杂度此前未知。我们的算法是组合型的，可视为一种原始-对偶算法。它通过搜索有向树及其对偶证书（一个边集子集链）来证明其流行性。事实上，我们的算法解决了两拟阵交中更一般的流行公共基问题，其中一个拟阵是由任意划分 $E = \bigcup_{v\in V} \delta(v)$ 定义的分割拟阵，另一个是 $E$ 上秩为 $|V|$ 的任意拟阵，且每个 $v \in V$ 对 $\delta(v)$ 中的元素具有偏序。我们将算法推广到含强制边或禁止边的情形。我们还研究了相关的流行彩色森林（更一般地，流行公共独立集）问题，其中边被划分为颜色类，任务是在所有彩色森林中寻找流行的彩色森林。对于弱排序情形，我们刻画了流行彩色森林多面体，从而证明可高效计算最小代价流行彩色森林。相比之下，我们证明即使排序是严格的，计算最小代价流行有向树仍是NP难的。