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Dense Steiner problems: Approximation algorithms and inapproximability

The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set $T$ of terminals in a graph $G$ by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the $δ$-dense version of {\sc Steiner Tree}, where each terminal has at least $δ|V(G)\setminus T|$ neighbours outside $T$, for a fixed $δ> 0$. They gave a PTAS for this problem. We study a generalization of pairwise $δ$-dense {\sc Steiner Forest}, which asks for a minimum-size forest in $G$ in which the nodes in each terminal set $T_1,\dots,T_k$ are connected, and every terminal in $T_i$ has at least $δ|T_j|$ neighbours in $T_j$, and at least $δ|S|$ nodes in $S = V(G)\setminus (T_1\cup\dots\cup T_k)$, for each $i, j$ in $\{1,\dots, k\}$ with $i\neq j$. Our first result is a polynomial-time approximation scheme for all $δ> 1/2$. Then, we show a $(\frac{13}{12}+\varepsilon)$-approximation algorithm for $δ= 1/2$ and any $\varepsilon > 0$. We also consider the $δ$-dense Group Steiner Tree problem as defined by Hauptmann and show that the problem is $\mathsf{APX}$-hard.