Approximation algorithms for solving the heterogeneous rooted tree/path cover problems

Abstract

In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph \(G=(V,E; w,f; r)\) and k construction teams, having nonuniform construction speeds \(\lambda _{1}\), \(\lambda _{2}\), \(\ldots \), \(\lambda _{k}\), where \(r\in V\) is a fixed common root, \(w:E\rightarrow {\mathbb {R}}^{+}\) is an edge-weight function, satisfying the triangle inequality, and \(f:V\rightarrow {\mathbb {R}}^{+}_{0}\) (i.e., \({\mathbb {R}}^{+}\cup \{0\})\) is a vertex-weight function with \(f(r)=0\), we are asked to find k trees for these k construction teams, each tree having the same root r, and collectively covering all vertices in V, the objective is to minimize the maximum completion time of k construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting k paths for k trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant \(\delta >0\), we first design a \(58.3286(1+\delta )\)-approximation algorithm to solve the HRTC problem, and this algorithm runs in time \(O(n^{2}(n+\frac{\log n}{\delta })+\log (w(E)+f(V)))\). Meanwhile, we present a simple \(116.6572(1+\delta )\)-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a \(\max \{2\rho , 2+\rho -\frac{2}{k}\}\)-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time \(O(n^{2})\), where \(\rho \) is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding \(\max \{2\rho , 2+\rho -\frac{2}{k}\}\)-approximation algorithm also resolves the HRPC problem.