A parallel algorithm for minimum weight set cover with small neighborhood property

Abstract

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 $\tau (1+3\epsilon )$ in $O(L{\log }_{1+\epsilon }\frac{n^3}{{\epsilon }^2}+4{\tau }^3{2}^{\tau }{L}^2\log n)$ rounds, where $0<\epsilon <\frac{1}{2}$ 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.}.