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SIAM J. Comput. 36, pp. 543-561 (19 pages)
The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals
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Add to FacebookWe consider the Directed Steiner Network problem, also called the Point‐to‐Point Connection problem. Given a directed graph $G$ and $p$ pairs $\{ (s_1,t_1), \dotsc, (s_p,t_p) \}$ of nodes in the graph, one has to find the smallest subgraph $H$ of $G$ that contains paths from $s_i$ to $t_i$ for all $i$. The problem is NP‐hard for general $p$, since the Directed Steiner Tree problem is a special case. Until now, the complexity was unknown for constant $p \geq 3$. We prove that the problem is polynomially solvable if $p$ is any constant number, even if nodes and edges in $G$ are weighted and the goal is to minimize the total weight of the subgraph $H$. In addition, we give an efficient algorithm for the Strongly Connected Steiner Subgraph problem for any constant $p$, where given a directed graph and $p$ nodes in the graph, one has to compute the smallest strongly connected subgraph containing the $p$ nodes.
© 2006 Society for Industrial and Applied Mathematics
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Received February 24, 2004
Accepted January 05, 2006
Published online August 07, 2006
Accepted January 05, 2006
Published online August 07, 2006
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