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29 March 2024
 
  » arxiv » 1604.5103

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On Directed Steiner Trees with Multiple Roots
Ondřej Suchý ;
Date 18 Apr 2016
AbstractWe introduce a new Steiner-type problem for directed graphs named extsc{$q$-Root Steiner Tree}. Here one is given a directed graph $G=(V,A)$ and two subsets of its vertices, $R$ of size $q$ and $T$, and the task is to find a minimum size subgraph of $G$ that contains a path from each vertex of $R$ to each vertex of $T$. The special case of this problem with $q=1$ is the well known extsc{Directed Steiner Tree} problem, while the special case with $T=R$ is the extsc{Strongly Connected Steiner Subgraph} problem.
We first show that the problem is W[1]-hard with respect to $|T|$ for any $q ge 2$. Then we restrict ourselves to instances with $R subseteq T$. Generalizing the methods of Feldman and Ruhl [SIAM J. Comput. 2006], we present an algorithm for this restriction with running time $O(2^{2q+4|T|}cdot n^{2q+O(1)})$, i.e., this restriction is FPT with respect to $|T|$ for any constant $q$. We further show that we can, without significantly affecting the achievable running time, loosen the restriction to only requiring that in the solution there are a vertex $v$ and a path from each vertex of $R$ to $v$ and from $v$ to each vertex of~$T$.
Finally, we use the methods of Chitnis et al. [SODA 2014] to show that the restricted version can be solved in planar graphs in $O(2^{O(q log q+|T|log q)}cdot n^{O(sqrt{q})})$ time.
Source arXiv, 1604.5103
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