A New Approach for Approximating Directed Rooted Networks
<|reference_start|>A New Approach for Approximating Directed Rooted Networks: We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are $k$ edge-disjoint $rt$-paths for every terminal $t\in T$. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for $k$-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of $\Omega (|T|/\log |T|)$ under $NP\neq ZPP$. Inspired by modern day applications, we focus on developing efficient algorithms for $k$-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for $k$-DST on such graphs, in which the approximation ratio depends (primarily) on the size of $S$. We present a randomized algorithm that finds a solution of weight at most $\mathcal O(k|S|\log |T|)$ times the optimal weight, and with high probability runs in polynomial time.<|reference_end|>
@article{cohen2024a,
    title={A New Approach for Approximating Directed Rooted Networks},
    author={Sarel Cohen, Lior Kamma, Aikaterini Niklanovits},
    journal={arXiv preprint arXiv:2407.07543},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.07543},
    primaryClass={cs.DS}
}