Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

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, $\text{textsc}Locating-DominatingSet$ (respectively, $\text{textsc}TestCover$) 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, $2^{2^{o(tw)}}\cdot poly(\mathrm{\mid }U\mathrm{\mid }+\mathrm{\mid }\mathcal{F}\mathrm{\mid }))$. 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 $\text{textsc}Locating-DominatingSet$ 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 Ð fails. To the best of our knowledge, $\text{textsc}Locating-DominatingSet$ 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 $\text{textsc}TestCover$ does not admit an algorithm running in time $2^{2^{o(k)}}\cdot poly(\mathrm{\mid }U\mathrm{\mid }+\mathrm{\mid }\mathcal{F}\mathrm{\mid })$. 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.