最优分布覆盖算法
Optimal Distributed Covering Algorithms
摘要：
我们提出了一个时间最优的确定性分布式算法近似的最小重量顶点覆盖秩$f$超图。该问题等价于最小权集覆盖问题，其中每个元素的频率由$f$限定。我们的算法的近似因子是$(f+ε)$。我们的算法在CONGEST模型中运行，并且对于常量$ε\in(0,1]$和$f\in N^+$需要$O(\logΔ/ \log\logΔ)$轮。这是该问题的第一个分布式算法，其运行时间不依赖于顶点的权重，也不依赖于顶点的数量。对于$f$和$ε$的常数值，我们的算法改进了KMW 06的$(f+ε)$ -近似算法，其运行时间是$O(\log Δ+ \log W)$，其中$W$是图中最大和最小顶点权重之间的比率。我们的算法还实现了$f$ -近似的问题在$O(f\log n)$轮，提高了经典的结果KVY 94，实现了运行时间的$O(f\log^2 n)$。最后，对于加权顶点覆盖（$f=2$），我们的算法实现了$O(\log n)$的\emph{deterministic}运行时间，匹配\emph{randomized}以前最好的结果KY 11。我们还表明，整数覆盖程序可以减少到最小重量集覆盖问题的分布式设置。这允许我们在$O(\frac{\logΔ}{\log\logΔ}+(f\cdot\log M)^{1.01}\logε^{-1}(\logΔ)^{0.01})$轮中实现$(f+ε)$ -近似积分解，其中$f$限制约束中的变量的数量，$Δ$限制变量出现的约束的数量，以及$M=\max \{1, 1/a_{\min}\}$，其中$a_{min}$是最小的归一化约束系数。这改进了KMW 06在积分情况下的结果，积分情况在$O(ε^{-4}\cdot f^4\cdot \log f\cdot\log(M\cdotΔ))$轮中运行。
英文摘要：
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 $(f+ε)$. Our algorithm runs in the CONGEST model and requires $O(\logΔ/ \log\logΔ)$ rounds, for constants $ε\in(0,1]$ and $f\in N^+$. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. For constant values of $f$ and $ε$, our algorithm improves over the $(f+ε)$-approximation algorithm of KMW06 whose running time is $O(\log Δ+ \log W)$, 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 KVY94 that achieves a running time of $O(f\log^2 n)$. Finally, for weighted vertex cover ($f=2$) our algorithm achieves a \emph{deterministic} running time of $O(\log n)$, matching the \emph{randomized} previously best result of KY11. 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 $(f+ε)$-approximate integral solution in $O(\frac{\logΔ}{\log\logΔ}+(f\cdot\log M)^{1.01}\logε^{-1}(\logΔ)^{0.01})$ rounds, where $f$ bounds the number of variables in a constraint, $Δ$ bounds the number of constraints a variable appears in, and $M=\max \{1, 1/a_{\min}\}$, where $a_{min}$ is the smallest normalized constraint coefficient. This improves over the results of KMW06 for the integral case, which runs in $O(ε^{-4}\cdot f^4\cdot \log f\cdot\log(M\cdotΔ))$ rounds.
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