## Abstract

In the k-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of k terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than O(k) is known. We show that k-Steiner Orientation is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form (log k)
^{o}(1) for FPT algorithms (assuming FPT =6 W[1]) and (log n)
^{o}
^{(1)} for purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). This constitutes a novel inapproximability result for polynomial-time algorithms obtained via tools from the FPT theory. Moreover, we prove k-Steiner Orientation to belong to W[1], which entails W[1]-completeness of (log k)
^{o}
^{(1)}-approximation for k-Steiner Orientation. This provides an example of a natural approximation task that is complete in a parameterized complexity class. Finally, we apply our technique to the maximization version of directed multicut - Max (k, p)Directed Multicut - where we are given a directed graph, k terminals pairs, and a budget p. The goal is to maximize the number of separated terminal pairs by removing p edges. We present a simple proof that the problem admits no FPT approximation with factor O(k
^{12 −ε}) (assuming FPT =6 W[1]) and no polynomial-time approximation with ratio O(|E(G)|
^{12 −ε}) (assuming NP 6⊆ co-RP).

Original language | English |
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Title of host publication | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |

Editors | Artur Czumaj, Anuj Dawar, Emanuela Merelli |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Pages | 104:1-104:19 |

Number of pages | 19 |

ISBN (Electronic) | 9783959771382 |

DOIs | |

Publication status | Published - 1 Jun 2020 |

Event | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany Duration: 8 Jul 2020 → 11 Jul 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 168 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |
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Country/Territory | Germany |

City | Virtual, Online |

Period | 8/07/20 → 11/07/20 |

### Bibliographical note

DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.## Keywords

- Approximation algorithms
- Fixed-parameter tractability
- Gap amplification
- Hardness of approximation