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

Let G = (V, E, w, rho, X) be a weighted undirected connected graph, where V is the set of vertices, E is the set of edges, X subset of V is a subset of terminals, w(e) > 0, for all e is an element of E denotes the weight associated with edge e, and rho(v) > 0, for all v is an element of V denotes the weight associated with vertex v. Let T be a Steiner tree in G to interconnect all terminals in X. For any two terminals, t', t '' is an element of X, we consider the weighted tree distance on T from t' to t '', defined as the weight of t '' times the classic tree distance on T from t' to t ''. The longest weighted tree distance on T between terminals is named the weighted diameter of T. The Minimum Diameter Vertex-Weighted Steiner Tree Problem (MDWSTP) asks for a Steiner tree in G of the minimum weighted diameter to interconnect all terminals in X. In this paper, we introduce two classes of parameterized graphs (PG), < X, mu >-PG and (X, lambda)-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 < X, mu >-PG, we present an approximation algorithm of a parameterized factor mu+1/2. For the MDWSTP in a vertex-weighted (X, lambda)-PG, we first present a simple approximation algorithm of a parameterized factor lambda, where lambda is tight when lambda >= 2, and further develop another approximation algorithm of a slightly improved factor.