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 $o(n^2)$.

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CONGEST 模型中的近似二分顶点覆盖

我们为 CONGEST 模型中二部图中的最小顶点覆盖问题给出了有效的分布式算法。根据 K\H{o}nig 定理，众所周知，在二部图中，最小顶点覆盖的大小等于最大匹配的大小。我们首先表明，与现有的 $O(n\log n)$-round 算法一起计算最大匹配，K\H{o}nig 定理的构造证明直接导致确定性 $O(n\log n )$-round 计算最小顶点覆盖的 CONGEST 算法。然后我们证明，通过调整构造，我们还可以将 \emph{approximate} 最大匹配转换为 \emph{approximate} 最小顶点覆盖。给定某个 $\delta>1$ 的 $(1-\delta)$-近似匹配，我们表明 $(1+O(\delta))$-近似顶点覆盖可以在 $O(D+\mathrm{poly}(\frac{\log n}{\delta}))$ 时间计算，其中$D$ 是图形的直径。当与已知的图聚类技术结合时，对于任何 $\varepsilon\in(0,1]$，这会导致 $\mathrm{poly}(\frac{\log n}{\varepsilon})$-time 确定性和还有一个稍微更快和更简单的随机 $O(\frac{\log n}{\varepsilon^3})$-round CONGEST 算法，用于计算二部图中的 $(1+\varepsilon)$-近似顶点覆盖。对于常数 $\varepsilon$，随机时间复杂度匹配 $\Omega(\log n)$ 下界，用于计算二部图中的 $(1+\varepsilon)$-近似顶点覆盖，即使在 LOCAL 模型中。我们的结果是也与一般图表中的情况相反，