Minimum Diameter Vertex-Weighted Steiner Tree

Abstract

Let $G = (V, E, w, \rho , \mathcal {T})$ be a weighted connected graph, where V is the vertex set, E is the edge set, $\mathcal {T} \subseteq V$ is a terminal subset, $w: E \rightarrow \mathbb {R}^{+}$ is an edge-weight function and $\rho : V \rightarrow \mathbb {R}^{+}$ is a vertex-weight function. The weighted diameter of a Steiner tree T in G spanning $\mathcal {T}$ is referred to as the longest weighted tree distance on T between terminals. The objective of the Minimum Diameter Vertex-Weighted Steiner Tree Problem (MDWSTP) is to construct a Steiner tree in G spanning $\mathcal {T}$ to minimize the weighted diameter.

In this paper, we study the MDWSTP in two classes of parameterized graphs, $\langle \mathcal {T}, \mu \rangle$-PG and $( \mathcal {T}, \lambda )$-PG, which are introduced from the perspective of the parameterized upper bound on the ratio of two vertex-weights, and a weaker version of the parameterized triangle inequality, respectively, and achieve simple approximation algorithms. For the MDWSTP in edge-weighted $\langle \mathcal {T}, \mu \rangle$-PG, we obtain a $\frac{\mu + 1}{2}$-factor approximation algorithm where $\frac{\mu + 1}{2}$ is tight. For the MDWSTP in vertex-weighted $( \mathcal {T}, \lambda )$-PG, we first obtain a $\lambda$-factor approximation algorithm where $\lambda$ is tight, and then develop a slightly improved approximation algorithm.