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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2020.104
URN: urn:nbn:de:0030-drops-125110
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12511/
Włodarczyk, Michał
Parameterized Inapproximability for Steiner Orientation by Gap Amplification
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 ?(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 ?(k^(1/2 - ε)) (assuming FPT ≠ W[1]) and no polynomial-time approximation with ratio ?(|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}
}
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Keywords: |
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approximation algorithms, fixed-parameter tractability, hardness of approximation, gap amplification |
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Collection: |
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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29.06.2020 |
