Arborescences, Colorful Forests, and Popularity

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.