We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is \((f+\varepsilon )\). Let \(\varDelta \) denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires \(O(\log {\varDelta } / \log \log \varDelta )\) rounds, for constants \(\varepsilon \in (0,1]\) and \(f\in {\mathbb {N}}^+\). This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and \(\varepsilon \), our algorithm improves over the \((f+\varepsilon )\)-approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is \(O(\log \varDelta + \log W)\), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in \(O(f\log n)\) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of \(O(f\log ^2 n)\). Finally, for weighted vertex cover (\(f=2\)) our algorithm achieves a deterministic running time of \(O(\log n)\), matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an \((f\lceil \log _2(M)+1 \rceil +\varepsilon )\)-approximate integral solution in

rounds, where f bounds the number of variables in a constraint, \(\varDelta \) bounds the number of constraints a variable appears in, and \(M=\max \left\{ 1, \lceil 1/a_{\min } \rceil \right\} \), where \(a_{\min }\) is the smallest normalized constraint coefficient.

我们提出了一种时间最优的确定性分布式算法，用于逼近等级为f的超图中的最小权重顶点覆盖。此问题等效于最小权重集覆盖问题，其中每个元素的频率均受f限制。我们算法的近似因子为\（（f + \ varepsilon）\）。令\（\ varDelta \）表示超图的最大程度。我们在算法运行拥塞模型，并且需要\（O（\日志{\ varDelta} / \日志\日志\ varDelta）\）轮，为常量\（\ varepsilon \在（0,1] \）和\（F \ in {\ mathbb {N}} ^ + \）。这是针对该问题的第一种分布式算法，其运行时间不取决于顶点权重或顶点数。因此，将另一个成员添加到可证明的最佳分布式算法的专有系列中。对于f和\（\ varepsilon \）的恒定值，我们的算法比Kuhn等人的\（（f + \ varepsilon）\） -逼近算法有所改进。（SODA，2006），其运行时间为\（O（\ log \ varDelta + \ log W）\），其中W是图中最大和最小顶点权重之间的比率。我们的算法还针对\（O（f \ log n）\）中的问题获得了f逼近改进了Khuller等人的经典结果。（J Algorithms，1994）实现运行时间为\（O（f \ log ^ 2 n）\）。最后，对于加权顶点覆盖（\（f = 2 \）），我们的算法获得了确定的运行时间\（O（\ log n）\），与Koufogiannakis和Young的随机化先前最佳结果相匹配（Distrib Comput，2011） 。我们还表明，在分布式设置中，整数覆盖程序可以减少到“最小重量设置覆盖率”问题。这使我们可以实现\（（f \ lceil \ log _2（M）+1 \ rceil + \ varepsilon）\）的近似积分解

回合，其中f限制约束中变量的数量，\（\ varDelta \）限制变量出现在其中的约束数量，并且\（M = \ max \ left \ {1，\ lceil 1 / a _ {\ min } \ rceil \ right \} \），其中\（a _ {\ min} \）是最小的归一化约束系数。