On the Steiner Tree $\frac{3}{2}$-Approximation for Quasi-Bipartite Graphs

Romeo Rizzi

November 1999


Let $G=(V,E)$ be an undirected simple graph and $w:E\mapsto
{\rm I\!R}_+$ be a non-negative weighting of the edges of $G$. Assume $V$ is partitioned as $R\cup X$. A Steiner tree is any tree $T$ of $G$ such that every node in $R$ is incident with at least one edge of $T$. The metric Steiner tree problem asks for a Steiner tree of minimum weight, given that $w$ is a metric. When $X$ is a stable set of $G$, then $(G,R,X)$ is called quasi-bipartite. In a SODA '99 paper, Rajagopalan and Vazirani introduced the notion of quasi-bipartiteness and gave a $(\frac{3}{2}+\epsilon)$ approximation algorithm for the metric Steiner tree problem, when $(G,R,X)$ is quasi-bipartite. In this paper, we simplify and strengthen the result of Rajagopalan and Vazirani. We also show how classical bit scaling techniques can be adapted to the design of approximation algorithms

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