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Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification, and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems such as min dominating set, min feedback vertex set, min set cover, min feedback arc set, and others. [ABSTRACT FROM AUTHOR]