Parameterizations of hitting set of bundles and inverse scope

Hitting Set of Bundles generalizes the ordinary Hitting Set problem in the way that prescribed bundles of elements rather than single elements have to be put in a hitting set. The goal is to minimize the total number of distinct elements in the solution. First we prove that Hitting Set of Bundles, with the number of hyperedges and the solution size as parameter, is W[1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W[1]$$\end{document}-complete. This contrasts to the to the corresponding parameterized Hitting Set version which is in FPT. Then we use this result to prove W[i]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W[i]$$\end{document}-hardness also for the Inverse Scope problem and some of its variants. This problem asks to identify small sets of chemical reactants being able to produce a given set of target compounds in a network of reactions. The problem has a graph-theoretic formulation as a reachability problem in directed graphs. On the positive side, we give an FPT algorithm where the parameter is the total number of compounds involved in the reactions.