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, among other things, the following three (tight) conditional lower bounds. \begin{enumerate} \item \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. \end{enumerate} 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. \begin{enumerate}[resume] \item \textsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| + |\calF|)$. \end{enumerate} After \textsc{Edge Clique Cover} and \textsc{BiClique Cover}, this is the only example that we know of that admits a double exponential lower bound when parameterized by the solution size. \begin{enumerate}[resume] \item \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, $2^{2^{o(\tw)}} \cdot poly(|U| + |\calF|))$. \end{enumerate} This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. We also present algorithms whose running times match the above lower bounds. We also investigate the parameterizations by several other structural graph parameters, answering some open problems from the literature.

翻译：我们研究了图与集合系统中识别问题的细粒度算法方面，重点关注定位控制集与测试覆盖问题。我们证明了以下三个（紧）条件性下界。\begin{enumerate} \item \textsc{定位控制集}在参数化解大小的情况下，不存在时间复杂度为$2^{o(k^2)} \cdot poly(n)$的算法，也不存在多项式时间核化算法能将解规模缩减并输出包含$2^{o(k)}$个顶点的核，除非\ETH\失效。\end{enumerate}据我们所知，\textsc{定位控制集}是第一个在参数化解大小时具有此类算法下界（指数中出现二次函数）的问题。\begin{enumerate}[resume] \item \textsc{测试覆盖}不存在时间复杂度为$2^{2^{o(k)}} \cdot poly(|U| + |\calF|)$的算法。\end{enumerate}继\textsc{边团覆盖}与\textsc{双团覆盖}之后，这是我们所知的第二个在参数化解大小时具有双指数下界的例子。\begin{enumerate}[resume] \item 以输入图（或自然辅助图）的树宽为参数时，\textsc{定位控制集}（或\textsc{测试覆盖}）不存在时间复杂度为$2^{2^{o(\tw)}} \cdot poly(n)$（或$2^{2^{o(\tw)}} \cdot poly(|U| + |\calF|))$的算法。\end{enumerate}该结果扩充了以树宽为参数化时具有双指数下界的NP完全问题的小型列表。我们还提出了运行时间与上述下界匹配的算法，并研究了多个其他结构图参数的参数化问题，解答了文献中的一些开放性问题。