## Abstract

The DIRECTED STEINER NETWORK (DSN) problem takes as input a directed edge-weighted graph G = (V, E) and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N ⊆ G for which there is an s → t path for each (s,t) ∈ D. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the BI-DSN_{PLANAR} problem, the aim is to compute a planar optimum solution N ⊆ G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of BI-DSN_{PLANAR}, unless FPT=W[1]. Additionally we study several generalizations of BI-DSNP_{LANAR} and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem, for which the solution network N ⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2 - ϵ)-approximation algorithm parameterized by k (for any ϵ > 0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k.

Original language | English |
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Title of host publication | 26th European Symposium on Algorithms, (ESA 2018). |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl |

Number of pages | 16 |

ISBN (Print) | 9783959770811 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 112 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 26th European Symposium on Algorithms, ESA 2018 |
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Country/Territory | Finland |

City | Helsinki |

Period | 20/08/18 → 22/08/18 |

## Keywords

- Bidirected graphs
- Directed steiner network
- Parameterized approximations
- Planar graphs
- Strongly connected steiner subgraph

## ASJC Scopus subject areas

- Software