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URN: urn:nbn:de:0030-drops-125110
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Parameterized Inapproximability for Steiner Orientation by Gap Amplification

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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 ≠ 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^(1/2 - ε)) (assuming FPT ≠ W[1]) and no polynomial-time approximation with ratio 𝒪(|E(G)|^(1/2 - ε)) (assuming NP ⊈ co-RP).

BibTeX - Entry

@InProceedings{wodarczyk:LIPIcs:2020:12511,
  author =	{Michał Włodarczyk},
  title =	{{Parameterized Inapproximability for Steiner Orientation by Gap Amplification}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{104:1--104:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12511},
  URN =		{urn:nbn:de:0030-drops-125110},
  doi =		{10.4230/LIPIcs.ICALP.2020.104},
  annote =	{Keywords: approximation algorithms, fixed-parameter tractability, hardness of approximation, gap amplification}
}

Keywords: approximation algorithms, fixed-parameter tractability, hardness of approximation, gap amplification
Seminar: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue date: 2020
Date of publication: 29.06.2020


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