The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals

Abstract

We consider the DIRECTED STEINER NETWORK problem, or the POINT-TO-POINT CONNECTION problem. Given a directed graph G and p pairs {(si, ti), .. ., (s,, tp)} of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from si to tj for all i.

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

“…▯ The (0-)DSN algorithm developed by Feldman and Ruhl [16] uses an algorithm for (0-)SCSS as a subprocedure. Using the algorithms developed in the proofs of Theorems 4.1 and 4.2 in case of arc-weight ratios 2 and 1, respectively, as the subprocedure, the running time of the overall algorithm of Feldman and Ruhl can be significantly improved by roughly halving the degree of its running time polynomial for the relevant case of these small arc-weight ratios.…”

“…The (0-)DSN algorithm of Feldman and Ruhl [16] is based on a simulation of a game in which one moves some tokens along the arcs of the input digraph, the price of the move being the total weight of the arcs used. To this end, they construct a so-called "game graph," with vertices representing the possible token positions and arcs representing the legal moves.…”

“…Conclusion. We extended previous work on directed Steiner problems [16]. In particular, we examined the impact of the ratio of the arc weights on the parameterized complexity of three Steiner problems with respect to the considered parameterizations.…”

“…Moreover, since the set cover problem is W[2]-complete [11] and it can also be formulated as a special case of both 0-DST and 0-SCSS [20], it follows that 0-DST, 0-SCSS, and 0-DSN are W[2]-hard with respect to the parameter p∕ min W . Finally, Feldman and Ruhl [16] showed that 0-DSN can be solved in Oðmn 4l−2 þ n 4l−1 log nÞ time using their Oðmn 2l−3 þ n 2l−2 log nÞ-time algorithm for 0-SCSS as a subprocedure. These algorithmic results directly lead to the question whether there are polynomial-time algorithms whose polynomial degree is independent of l. In other words, Feldman and Ruhl asked for the fixed-parameter tractability of (0-)DSN and (0-)SCSS with respect to the parameter l.…”

Energy efficient communication in ad hoc networks from user's and designer's perspective

“…We assume that different transmissions can be aggregated if they pass through the same edge. We use the algorithm provided by Feldman and Ruhl [15] for the p-Directed Steiner Network problem. In this problem, we are given a directed weighted graph …”