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.

翻译：本文研究了图与集合系统中识别问题的精细算法特性，重点关注定位控制集与测试覆盖问题。我们证明了这些问题在基于树宽与解规模作为参数时的（紧致）条件性下界。具体而言，以输入图（或自然辅助图）的树宽为参数的 \textsc{定位控制集}（相应地，\textsc{测试覆盖}）问题不存在运行时间为 $2^{2^{o(tw)}} \cdot poly(n)$（相应地，$2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|)$）的算法。这一结果扩充了在树宽参数化下具有双重指数下界的 NP 完全问题列表。其次，我们首先证明除非 \ETH\ 不成立，否则 \textsc{定位控制集} 不存在运行时间为 $2^{o(k^2)} \cdot poly(n)$ 的算法，也不存在能在多项式时间内将解规模缩减并输出具有 $2^{o(k)}$ 个顶点的核的多项式时间核化算法。据我们所知，\textsc{定位控制集} 是首个在基于解规模参数化时具有此类算法下界（指数部分为二次函数）的问题。最后，我们证明 \textsc{测试覆盖} 不存在运行时间为 $2^{2^{o(k)}} \cdot poly(|U| + |\mathcal{F}|)$ 的算法。这也是在基于解规模参数化时具有双重指数下界的少数问题之一。我们还提出了运行时间与上述下界匹配的算法。