From Trees to Binary Trees: Approximation Algorithms and a PTAS

Kellerhals, Krebs, Nichterlein, and Schmid introduced the optimization problem Tree in Binary Tree (TBT), where, given a tree $T$, the task is to find a binary tree $S^*$ with $V(T)=V(S^*)$ that minimizes $\sum_{uv\in E(T)} \operatorname{dist}_{S^*}(u,v)$. They also introduced the variant Tree in Binary Tree With Steiner Nodes (TBT-SN), where the condition $V(T)=V(S^*)$ is relaxed to $V(T)\subseteq V(S^*)$. We give a simple linear-time $\frac{8}{3}$-approximation algorithm for TBT and a polynomial-time approximation scheme (PTAS) for TBT-SN. Our main building idea in both algorithms is to iteratively consider subtrees of the tree $T$ which contain all vertices below some vertex $v$ and construct a solution where $v$ has degree 1. For the PTAS, we cut $T$ into smaller trees, solve these instances, modify the obtained solutions, and combine them to achieve a solution for the original instance.