This paper studies the minimum weight set cover (MinWSC) problem with a {\em small neighborhood cover} (SNC) property proposed by Agarwal {\it et al.} in \cite{Agarwal.}. A parallel algorithm for MinWSC with $\tau$-SNC property is presented, obtaining approximation ratio $\tau(1+3\varepsilon)$ in $O(L\log_{1+\varepsilon}\frac{n^3}{\varepsilon^2}+ 4\tau^{3}2^\tau L^2\log n)$ rounds, where $0< \varepsilon <\frac{1}{2}$ is a constant, $n$ is the number of elements, and $L$ is a parameter related to SNC property. Our results not only improve the approximation ratio obtained in \cite{Agarwal.}, but also answer two questions proposed in \cite{Agarwal.}.
翻译：本文研究了Agarwal 等公司在\ cite{Agarwal.} 中提议的“小街区覆盖” (SNC) 财产的最低重量(MINWSC) 问题。 提出了“ 最低重量( MINWSC) ” 与 美元- SNC 财产的平行算法, 以 $( log} 1+3 varepsilón} frac} (n) 3\\ varepsilon} 4\ tau} 2 ⁇ 2 ⁇ Tau L ⁇ 2\ log n) 回合, 其中 $( varepsilon) {frac} {1\\\ 2} 是固定的, $( $) 是元素的数量, $( $) 是与 SNC 财产有关的参数。 我们的结果不仅改善了在\ cite{Agarwal.} 中提出的近似比率, 而且还回答了\ cite{Agarwal.} 中提出的两个问题 。
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