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On rooted $k$-connectivity problems in quasi-bipartite digraphs
arXiv - CS - Data Structures and Algorithms Pub Date : 2020-09-21 , DOI: arxiv-2009.10160
Zeev Nutov

We consider the directed Rooted Subset $k$-Edge-Connectivity problem: given a set $T \subseteq V$ of terminals in a digraph $G=(V+r,E)$ with edge costs and an integer $k$, find a min-cost subgraph of $G$ that contains $k$ edge disjoint $rt$-paths for all $t \in T$. The case when every edge of positive cost has head in $T$ admits a polynomial time algorithm due to Frank, and the case when all positive cost edges are incident to $r$ is equivalent to the $k$-Multicover problem. Recently, [Chan et al. APPROX20] obtained ratio $O(\ln k \ln |T|)$ for quasi-bipartite instances, when every edge in $G$ has an end in $T+r$. We give a simple proof for the same ratio for a more general problem of covering an arbitrary $T$-intersecting supermodular set function by a minimum cost edge set, and for the case when only every positive cost edge has an end in $T+r$.

中文翻译:

准二部有向图中的根 $k$-连通性问题

我们考虑有向根子集 $k$-Edge-Connectivity 问题:给定一个有向图 $G=(V+r,E)$ 中终端的集合 $T \subseteq V$,带有边成本和一个整数 $k$,找到 $G$ 的最小成本子图,其中包含所有 $t \in T$ 的 $k$ 边不相交 $rt$-paths。正成本的每条边都在 $T$ 中的情况承认由于 Frank 的多项式时间算法,并且所有正成本边都与 $r$ 相关的情况等价于 $k$-Multicover 问题。最近,[陈等人。APPROX20] 获得了准二分实例的比率 $O(\ln k \ln |T|)$,当 $G$ 中的每条边都在 $T+r$ 中结束时。对于通过最小成本边集覆盖任意 $T$ 相交超模集函数的更一般问题,我们给出了相同比率的简单证明,并且对于只有每个正成本边以 $T+ 结束的情况r$。
更新日期:2020-09-23
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