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arXiv 1802.08189cs.DM
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Steiner网络问题关于终端数的复杂性

Complexity of the Steiner Network Problem with Respect to the Number of Terminals

Eduard Eiben, Dušan Knop, Fahad Panolan, Ondřej Suchý

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摘要：

在有向Steiner网络问题中，我们给出了弧加权有向图$G$，一组终端$T \subseteq V(G)$，以及一个（未加权）有向请求图$R$和$V(R)=T$。我们的任务是输出一个最小代价的子图$G' \subseteq G$，使得对于所有的$st \in A(R)$，在$G'$中存在从$s$到$t$的有向路径。已知的是，该问题可以在时间$|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl，SIAM J. Comput. 2006]并且即使$G$是平面的也不能在时间$|V(G)|^{o(|A(R)|)}$中求解，除非指数时间假设（ETH）失败[Chitnis等人，SODA 2014]。然而，由于这种简化（以及显示问题难度的其他简化）仅表明除非ETH失败，否则问题无法及时解决$|V(G)|^{o(|T|)}$，因此在指数中相对于$|T|$的复杂性存在显著差距。我们证明了有向Steiner网络在时间$f(R)\cdot |V(G)|^{O(c_g \cdot |T|)}$中是可解的，其中$c_g$是一个常数，仅依赖于$G$的亏格，$f$是一个可计算函数.我们通过证明对于一般图上的问题的任何函数$f$都没有$f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)}$算法来补充这个结果，除非ETH失败。

英文摘要：

In the Directed Steiner Network problem we are given an arc-weighted digraph $G$, a set of terminals $T \subseteq V(G)$, and an (unweighted) directed request graph $R$ with $V(R)=T$. Our task is to output a subgraph $G' \subseteq G$ of the minimum cost such that there is a directed path from $s$ to $t$ in $G'$ for all $st \in A(R)$. It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time $|V(G)|^{o(|A(R)|)}$ even if $G$ is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time $|V(G)|^{o(|T|)}$ unless ETH fails, there is a significant gap in the complexity with respect to $|T|$ in the exponent. We show that Directed Steiner Network is solvable in time $f(R)\cdot |V(G)|^{O(c_g \cdot |T|)}$, where $c_g$ is a constant depending solely on the genus of $G$ and $f$ is a computable function. We complement this result by showing that there is no $f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)}$ algorithm for any function $f$ for the problem on general graphs, unless ETH fails.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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