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(see paper for full abstract)  Cut problems and connectivity problems on digraphs are two well-studiedclasses of problems from the viewpoint of parameterized complexity. After aseries of papers over the last decade, we now have (almost) tight bounds forthe running time of several standard variants of these problems parameterizedby two parameters: the number $k$ of terminals and the size $p$ of thesolution. When there is evidence of FPT intractability, then the next naturalalternative is to consider FPT approximations. In this paper, we show two typesof results for several directed cut and connectivity problems, building onexisting results from the literature: first is to circumvent the hardnessresults for these problems by designing FPT approximation algorithms, oralternatively strengthen the existing hardness results by creating"gap-instances" under stronger hypotheses such as the (Gap-)Exponential TimeHypothesis (ETH).

FPT Inapproximability of Directed Cut and Connectivity Problems