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We carry out a systematic study of a natural covering problem, used foridentification across several areas, in the realm of parameterized complexity.In the {\sc Test Cover} problem we are given a set $[n]=\{1,...,n\}$ of itemstogether with a collection, $\cal T$, of distinct subsets of these items calledtests. We assume that $\cal T$ is a test cover, i.e., for each pair of itemsthere is a test in $\cal T$ containing exactly one of these items. Theobjective is to find a minimum size subcollection of $\cal T$, which is still atest cover. The generic parameterized version of {\sc Test Cover} is denoted by$p(k,n,|{\cal T}|)$-{\sc Test Cover}. Here, we are given $([n],\cal{T})$ and apositive integer parameter $k$ as input and the objective is to decide whetherthere is a test cover of size at most $p(k,n,|{\cal T}|)$. We study fourparameterizations for {\sc Test Cover} and obtain the following:  (a) $k$-{\sc Test Cover}, and $(n-k)$-{\sc Test Cover} are fixed-parametertractable (FPT).  (b) $(|{\cal T}|-k)$-{\sc Test Cover} and $(\log n+k)$-{\sc Test Cover} areW[1]-hard. Thus, it is unlikely that these problems are FPT.

Parameterized Study of the Test Cover Problem