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Given a graph $G = (V,E)$ and a subset $T \subseteq V$ of terminals, a\emph{Steiner tree} of $G$ is a tree that spans $T$. In the vertex-weightedSteiner tree (VST) problem, each vertex is assigned a non-negative weight, andthe goal is to compute a minimum weight Steiner tree of $G$.  We study a natural generalization of the VST problem motivated by multi-levelgraph construction, the \emph{vertex-weighted grade-of-service Steiner treeproblem} (V-GSST), which can be stated as follows: given a graph $G$ andterminals $T$, where each terminal $v \in T$ requires a facility of a minimumgrade of service $R(v)\in \{1,2,\ldots\ell\}$, compute a Steiner tree $G'$ byinstalling facilities on a subset of vertices, such that any two verticesrequiring a certain grade of service are connected by a path in $G'$ with theminimum grade of service or better. Facilities of higher grade are more costlythan facilities of lower grade. Multi-level variants such as this one can beuseful in network design problems where vertices may require facilities ofvarying priority.  While similar problems have been studied in the edge-weighted case, they havenot been studied as well in the more general vertex-weighted case. We firstdescribe a simple heuristic for the V-GSST problem whose approximation ratiodepends on $\ell$, the number of grades of service. We then generalize thegreedy algorithm of [Klein \& Ravi, 1995] to show that the V-GSST problemadmits a $(2 \ln |T|)$-approximation, where $T$ is the set of terminalsrequiring some facility. This result is surprising, as it shows that the(seemingly harder) multi-grade problem can be approximated as well as the VSTproblem, and that the approximation ratio does not depend on the number ofgrades of service.

Approximation algorithms for the vertex-weighted grade-of-service Steiner tree problem