  We investigate fine-grained algorithmic aspects of identification problems in
graphs and set systems, with a focus on Locating-Dominating Set and Test Cover.
We prove the (tight) conditional lower bounds for these problems when
parameterized by treewidth and solution as. Formally,
\textsc{Locating-Dominating Set} (respectively, \textsc{Test Cover})
parameterized by the treewidth of the input graph (respectively, of the natural
auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}}
\cdot poly(n)$ (respectively, $2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|))$.
This result augments the small list of NP-Complete problems that admit double
exponential lower bounds when parameterized by treewidth. Then, we first prove
that \textsc{Locating-Dominating Set} does not admit an algorithm running in
time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization algorithm
that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices,
unless the \ETH\ fails. To the best of our knowledge,
\textsc{Locating-Dominating Set} is the first problem that admits such an
algorithmic lower-bound (with a quadratic function in the exponent) when
parameterized by the solution size. Finally, we prove that \textsc{Test Cover}
does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| +
|\mathcal{F}|)$. This is also a rare example of the problem that admits a
double exponential lower bound when parameterized by the solution size.
  We also present algorithms whose running times match the above lower bounds.
