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

TL;DR

This paper introduces a parallel algorithm for the minimum weight set cover problem with small neighborhood property, achieving improved approximation ratios and addressing open questions from prior research.

Contribution

It presents a new parallel algorithm for MinWSC with small neighborhood cover, improving approximation ratios and answering open questions from previous work.

Findings

Achieves approximation ratio of τ(1+3ε)

Operates in specific round complexity bounds

Improves upon previous approximation results

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 $\tau(1+3\varepsilon)$ 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.}.