Let G=(V,E) be an undirected graph with costs on the edges specified by a weighting w of the edges by positive reals. A Steiner tree is any tree of G which spans all nodes in a given subset R of V. When V / R is a stable set of G, then (G,R) is called quasi-bipartite. Rajagopalan and Vazirani introduced the notion of quasi-bipartiteness and showed that the Iterated 1-Steiner heuristic always produces a Steiner tree of total cost at most 3/2 the optimal when (G,R) is quasi-bipartite and w is a metric. In this paper, we give a more direct and much simpler proof of this result. Next, we show how a bit scaling approach yields a polynomial time implementation of the Iterated 1-Steiner heuristic. This gives a 3/2-approximation algorithm for the problem considered by Rajagopalan and Vazirani.

### On Rajagopalan and Vazirani's 3/2-Approximation Bound for the Iterated 1-Steiner Heuristic

#### Abstract

Let G=(V,E) be an undirected graph with costs on the edges specified by a weighting w of the edges by positive reals. A Steiner tree is any tree of G which spans all nodes in a given subset R of V. When V / R is a stable set of G, then (G,R) is called quasi-bipartite. Rajagopalan and Vazirani introduced the notion of quasi-bipartiteness and showed that the Iterated 1-Steiner heuristic always produces a Steiner tree of total cost at most 3/2 the optimal when (G,R) is quasi-bipartite and w is a metric. In this paper, we give a more direct and much simpler proof of this result. Next, we show how a bit scaling approach yields a polynomial time implementation of the Iterated 1-Steiner heuristic. This gives a 3/2-approximation algorithm for the problem considered by Rajagopalan and Vazirani.
##### Scheda breve Scheda completa Scheda completa (DC) Steiner tree; local search; Iterated 1-Steiner heuristic; bit scaling
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11562/409620.1`
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