A {\em bipartite tournament} is a directed graph $T:=(A \cup B, E)$ such that every pair of vertices $(a,b), a\in A,b\in B$ are connected by an arc, and no arc connects two vertices of $A$ or two vertices of $B$. A {\em feedback vertex set} is a set $S$ of vertices in $T$ such that $T - S$ is acyclic. In this article we consider the {\sc Feedback Vertex Set} problem in bipartite tournaments. Here the input is a bipartite tournament $T$ on $n$ vertices together with an integer $k$, and the task is to determine whether $T$ has a feedback vertex set of size at most $k$. We give a new algorithm for {\sc Feedback Vertex Set in Bipartite Tournaments}. The running time of our algorithm is upper-bounded by $O(1.6181^k + n^{O(1)})$, improving over the previously best known algorithm with running time $2^kk^{O(1)} + n^{O(1)}$ [Hsiao, ISAAC 2011]. As a by-product, we also obtain the fastest currently known exact exponential-time algorithm for the problem, with running time $O(1.3820^n)$.

翻译：{\em 二分图竞赛图}是一种有向图$T:=(A \cup B, E)$，其满足任意顶点对$(a,b), a\in A,b\in B$之间均存在一条弧相连，且$A$中任意两顶点或$B$中任意两顶点之间均不存在弧。{\em 反馈顶点集}是$T$中的一个顶点子集$S$，使得$T - S$是无环的。本文研究二分图竞赛图中的{\sc 反馈顶点集}问题。该问题的输入是一个包含$n$个顶点的二分图竞赛图$T$及整数$k$，任务是判断$T$是否存在大小不超过$k$的反馈顶点集。我们提出了一种求解{\sc 二分图竞赛图反馈顶点集}问题的新算法。该算法的时间复杂度上界为$O(1.6181^k + n^{O(1)})$，优于此前已知的最优算法时间复杂度$2^kk^{O(1)} + n^{O(1)}$[Hsiao, ISAAC 2011]。作为副产品，我们还得到了目前该问题最快的精确指数时间算法，其时间复杂度为$O(1.3820^n)$。