An Improved Approximation Ratio to the Partial-Terminal Steiner Tree Problem

Abstract

We consider a generalization of both the classic Steiner tree problem and the terminal Steiner tree problem. Given a complete graph <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${ G = (V,E)}$</tex> </formula> with a metric cost function <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${ c:E \rightarrow {\BBQ_ \geq }}$</tex> </formula> and two proper subsets <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ R \subset V$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ R^\prime \subseteq R$</tex> </formula> , a partial-terminal Steiner tree is a Steiner tree which contains all vertices in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\it R$</tex> </formula> such that all vertices in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$R^\prime$</tex> </formula> must be leaves. The partial-terminal Steiner tree problem is to find a partial-terminal Steiner tree of the minimum cost in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$G$</tex> </formula> . The previously best-known approximation ratio of the problem is <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\bf \rho$</tex> </formula> is the approximation ratio of the Steiner tree problem. In this paper, we improve the ratio from <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho$</tex> </formula> to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho - {\rho \over {3\rho - 2}} - f$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$f$</tex> </formula> is a non-negative function whose value is between 0 and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ \rho - {\rho \over {3\rho - 2}}$</tex> </formula> .