Parameterized Approximations for the Minimum Diameter Vertex-Weighted Steiner Tree Problem in Graphs with Parameterized Weights

Abstract

Let [Formula: see text] be a weighted undirected connected graph, where [Formula: see text] is the set of vertices, [Formula: see text] is the set of edges, [Formula: see text] is a subset of terminals, [Formula: see text] denotes the weight associated with edge [Formula: see text], and [Formula: see text] denotes the weight associated with vertex [Formula: see text]. Let [Formula: see text] be a Steiner tree in [Formula: see text] to interconnect all terminals in [Formula: see text]. For any two terminals, [Formula: see text], we consider the weighted tree distance on [Formula: see text] from [Formula: see text] to [Formula: see text], defined as the weight of [Formula: see text] times the classic tree distance on [Formula: see text] from [Formula: see text] to [Formula: see text]. The longest weighted tree distance on [Formula: see text] between terminals is named the weighted diameter of [Formula: see text]. The Minimum Diameter Vertex-Weighted Steiner Tree Problem (MDWSTP) asks for a Steiner tree in [Formula: see text] of the minimum weighted diameter to interconnect all terminals in [Formula: see text]. In this paper, we introduce two classes of parameterized graphs (PG), [Formula: see text]-PG and [Formula: see text]-PG, in terms of the parameterized upper bound on the ratio of two vertex weights, and a weaker version of the parameterized triangle inequality, respectively, and present approximation algorithms of a parameterized factor for the MDWSTP in them. For the MDWSTP in an edge-weighted [Formula: see text]-PG, we present an approximation algorithm of a parameterized factor [Formula: see text]. For the MDWSTP in a vertex-weighted [Formula: see text]-PG, we first present a simple approximation algorithm of a parameterized factor [Formula: see text], where [Formula: see text] is tight when [Formula: see text], and further develop another approximation algorithm of a slightly improved factor.