SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 546-574, January 2021.
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parameterization. In particular, on one hand Steiner Tree is known to be ${APX}$-hard, and ${W[2]}$-hard on the other, if parameterized by the number of nonterminals (Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (${EPAS}$), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (${PSAKS}$) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For none of these is an ${EPAS}$ likely to exist for the studied parameter. For Steiner Forest an easy observation shows that the problem is ${APX}$-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is ${W[1]}$-hard. Nevertheless, we show that an ${EPAS}$ exists for Unweighted Directed Steiner Tree, but a ${PSAKS}$ does not. We also prove that there is an ${EPAS}$ and a ${PSAKS}$ for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

中文翻译：

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具有少量 Steiner 顶点的 Steiner 树的参数化逼近方案

SIAM 离散数学杂志，第 35 卷，第 1 期，第 546-574 页，2021 年 1 月。
我们研究了 Steiner Tree 问题，其中需要在边加权图中以最便宜的方式连接一组终端顶点。从近似和参数化的角度对这个问题进行了广泛的研究。特别是，如果通过最优解中的非终结点（Steiner 顶点）的数量进行参数化，则一方面 Steiner 树已知为 ${APX}$-hard，另一方面为 ${W[2]}$-hard . 与此相反，我们给出了一个有效的参数化近似方案 (${EPAS}$)，它绕过了两种硬度结果。此外，我们的方法意味着所考虑的参数存在多项式大小近似核化方案（${PSAKS}$）。我们进一步研究了 Steiner Tree 的其他变体的参数化近似性，例如 Directed Steiner Tree 和 Steiner Forest。对于所研究的参数，这些都不可能存在 ${EPAS}$。对于 Steiner Forest，一个简单的观察表明问题是 ${APX}$-hard，即使输入图不包含 Steiner 顶点。对于有向斯坦纳树，我们证明在研究参数的任何函数内逼近是 ${W[1]}$-hard。尽管如此，我们证明了无权有向 Steiner 树存在 ${EPAS}$，但 ${PSAKS}$ 不存在。我们还证明，如果除了 Steiner 顶点的数量之外，将最优解的连通分量的数量视为参数，则 Steiner Forest 存在 ${EPAS}$ 和 ${PSAKS}$。对于有向斯坦纳树，我们证明在研究参数的任何函数内逼近是 ${W[1]}$-hard。尽管如此，我们证明了无权有向 Steiner 树存在 ${EPAS}$，但 ${PSAKS}$ 不存在。我们还证明，如果除了 Steiner 顶点的数量之外，将最优解的连通分量的数量视为参数，则 Steiner Forest 存在 ${EPAS}$ 和 ${PSAKS}$。对于有向斯坦纳树，我们证明在研究参数的任何函数内逼近是 ${W[1]}$-hard。尽管如此，我们证明了无权有向 Steiner 树存在 ${EPAS}$，但 ${PSAKS}$ 不存在。我们还证明，如果除了 Steiner 顶点的数量之外，将最优解的连通分量的数量视为参数，则 Steiner Forest 存在 ${EPAS}$ 和 ${PSAKS}$。