The Heterogeneous Rooted Tree Cover Problem

We consider the heterogeneous rooted tree cover (HRTC) problem. Concretely, given an undirected complete graph \(G=(V,E)\) with a root \(r\in V\) , an edge-weight function \(w:E\rightarrow R^{+}\) satisfying the triangle inequality, a vertex-weight function \(f:V{\setminus }\{r\}\rightarrow R^{+}_{0}\) , and k construction teams having nonuniform construction speeds \(\lambda _{1}\) , \(\lambda _{2}\) , \(\ldots \) , \(\lambda _{k}\) , we are asked to find k trees for these k construction teams to cover all vertices in V, each tree starting at the same root r, i.e., k trees having a sole common vertex called root r, the objective is to minimize the maximum completion time, where the completion time of each team is the total construction weight of its related tree divided by its construction speed.     In this paper, we first design a \(58.3286(1+\delta )\) -approximation algorithm to solve the HRTC problem in time \(O(n^{3}(1+\frac{1}{\delta })+\log (w(E)+f(V\backslash \{r\})))\) for any \(\delta >0\) . In addition, we present a \(\max \{2\rho , 2+\rho -\frac{2}{k}\}\) -approximation algorithm for resolving the HRTC problem in time \(O(n^{2})\) , where \(\rho \) is the ratio between the maximum and minimum speed of these k teams.