We give efficient distributed algorithms for the minimum vertex cover problem in bipartite graphs in the CONGEST model. From K\H{o}nig's theorem, it is well known that in bipartite graphs the size of a minimum vertex cover is equal to the size of a maximum matching. We first show that together with an existing $O(n\log n)$-round algorithm for computing a maximum matching, the constructive proof of K\H{o}nig's theorem directly leads to a deterministic $O(n\log n)$-round CONGEST algorithm for computing a minimum vertex cover. We then show that by adapting the construction, we can also convert an \emph{approximate} maximum matching into an \emph{approximate} minimum vertex cover. Given a $(1-\delta)$-approximate matching for some $\delta>1$, we show that a $(1+O(\delta))$-approximate vertex cover can be computed in time $O(D+\mathrm{poly}(\frac{\log n}{\delta}))$, where $D$ is the diameter of the graph. When combining with known graph clustering techniques, for any $\varepsilon\in(0,1]$, this leads to a $\mathrm{poly}(\frac{\log n}{\varepsilon})$-time deterministic and also to a slightly faster and simpler randomized $O(\frac{\log n}{\varepsilon^3})$-round CONGEST algorithm for computing a $(1+\varepsilon)$-approximate vertex cover in bipartite graphs. For constant $\varepsilon$, the randomized time complexity matches the $\Omega(\log n)$ lower bound for computing a $(1+\varepsilon)$-approximate vertex cover in bipartite graphs even in the LOCAL model. Our results are also in contrast to the situation in general graphs, where it is known that computing an optimal vertex cover requires $\tilde{\Omega}(n^2)$ rounds in the CONGEST model and where it is not even known how to compute any $(2-\varepsilon)$-approximation in time $o(n^2)$.

翻译：我们为最小的顶点配置了高效分布式算法, 在 CONEST 模型中, 最低的顶点覆盖了双面图中的问题。 从 K\ H{ o} nig 的方格中, 众所周知, 在双面图中, 最小的顶点覆盖的大小等于最大匹配的大小。 我们首先显示, 与现有的 $( n\ log n) 的全方位算法一起计算最大匹配, K\ h{ o} nig 的正文直接导致确定性 $( n\ log n) 的全方位 CONEST 计算最低的顶层盖子。 在 美元( 美元) 平面图中, 美元( 美元) 平面图中, 美元是已知的 美元 。