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Abstract
Consider 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 naturally extend to preferences over arborescences rooted at \(r\) . We present a polynomial-time algorithm that decides whether a given input instance admits a popular arborescence, i.e., one for which there is no “more popular” arborescence. In fact, our algorithm solves the more general popular common base problem in the intersection of two matroids: we are given an arbitrary matroid \(M=(E,\mathcal{I})\) and a partition matroid \(M_{\text{part}}\) over \(E\) , where partition classes correspond to a set \(V\) of agents with \(|V|={\rm rank}(M)\) and each agent has a partial order preference over its associated partition class; the problem asks for a common base of \(M\) and \(M_{\text{part}}\) such that there is no “more popular” common base. Our algorithm is combinatorial, and can be regarded as a primal–dual algorithm. It searches for a solution along with its dual certificate, a chain of subsets of \(E\) , witnessing its popularity. Our generalized results, expressed in terms of matroids, demonstrate that the identification of agents with vertices of the graph in the popular arborescence problem is not essential. We also study the related popular common independent set problem. For the case with weak rankings, we formulate the popular common independent set polytope, and thus show that a minimum-cost popular common independent set can be computed efficiently. By contrast, we prove that it is \(\mathsf{NP}\) -hard to compute a minimum-cost popular arborescence, even when rankings are strict.
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