Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911115159535616 |
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| author | Chakraborty, Dipayan Foucaud, Florent Majumdar, Diptapriyo Tale, Prafullkumar |
| author_facet | Chakraborty, Dipayan Foucaud, Florent Majumdar, Diptapriyo Tale, Prafullkumar |
| contents | 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 Ð 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover Chakraborty, Dipayan Foucaud, Florent Majumdar, Diptapriyo Tale, Prafullkumar Data Structures and Algorithms Computational Complexity Discrete Mathematics 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 Ð 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. |
| title | Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics |
| url | https://arxiv.org/abs/2402.08346 |