Abstract
Cut problems and connectivity problems on digraphs are two wellstudied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number k of terminals and the size p of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gapinstances" under stronger hypotheses such as the (Gap)Exponential Time Hypothesis (ETH). Formally, we show the following results:
Cutting paths between a set of terminal pairs, i.e., Directed Multicut: Pilipczuk and Wahlstrom [TOCT '18] showed that Directed Multicut is W[1]hard when parameterized by p if k=4. We complement this by showing the following two results:
 Directed Multicut has a k/2approximation in 2^{O(p^2)}* n^{O(1)} time (i.e., a 2approximation if k=4),
 Under GapETH, Directed Multicut does not admit an (59/58epsilon)approximation in f(p)* n^{O(1)} time, for any computable function f, even if k=4.
Connecting a set of terminal pairs, i.e., Directed Steiner Network (DSN): The DSN problem on general graphs is known to be W[1]hard parameterized by p+k due to Guo et al. [SIDMA '11]. Dinur and Manurangsi [ITCS '18] further showed that there is no FPT k^{1/4o(1)}approximation algorithm parameterized by k, under GapETH. Chitnis et al. [SODA '14] considered the restriction to special graph classes, but unfortunately this does not lead to FPT algorithms either: DSN on planar graphs is W[1]hard parameterized by k. In this paper we consider the DSN_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for DSN_Planar:
 DSN_Planar has no (2epsilon)approximation in FPT time parameterized by k, under GapETH. This answers in the negative a question of Chitnis et al. [ESA '18].
 DSN_Planar is W[1]hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)approximation for DSN_Planar in f(k,p,epsilon)* n^{o(k+sqrt{p+1/epsilon})} time for any computable function f.
Pairwise connecting a set of terminals, i.e., Strongly Connected Steiner Subgraph (SCSS): Guo et al. [SIDMA '11] showed that SCSS is W[1]hard parameterized by p+k, while Chitnis et al. [SODA '14] showed that SCSS remains W[1]hard parameterized by p, even if the input graph is planar. In this paper we consider the SCSS_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for SCSS_Planar:
 SCSS_Planar is W[1]hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)approximation for SCSS_Planar in f(k,p,epsilon)* n^{o(sqrt{k+p+1/epsilon})} time for any computable function f.
Previously, the only known FPT approximation results for SCSS applied to general graphs parameterized by k: a 2approximation by Chitnis et al. [IPEC '13], and a matching (2epsilon)hardness under GapETH by Chitnis et al. [ESA '18].
BibTeX  Entry
@InProceedings{chitnis_et_al:LIPIcs:2019:11469,
author = {Rajesh Chitnis and Andreas Emil Feldmann},
title = {{FPT Inapproximability of Directed Cut and Connectivity Problems}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {8:18:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771290},
ISSN = {18688969},
year = {2019},
volume = {148},
editor = {Bart M. P. Jansen and Jan Arne Telle},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2019/11469},
URN = {urn:nbn:de:0030drops114692},
doi = {10.4230/LIPIcs.IPEC.2019.8},
annote = {Keywords: Directed graphs, cuts, connectivity, Steiner problems, FPT inapproximability}
}
Keywords: 

Directed graphs, cuts, connectivity, Steiner problems, FPT inapproximability 
Collection: 

14th International Symposium on Parameterized and Exact Computation (IPEC 2019) 
Issue Date: 

2019 
Date of publication: 

04.12.2019 