Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover
| Autor: | Chakraborty, Dipayan, Foucaud, Florent, Majumdar, Diptapriyo, Tale, Prafullkumar |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Abstract shortened to fit the requirements. This version contains a subset of results mentioned in the previous version |
| Databáze: | arXiv |
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