Timeline for What are efficient pooling designs for RT-PCR tests?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2021 at 13:54 | comment | added | Benoît Kloeckner | Hi again, could you please contact me by email (I am easy to find and I don't know another way to get in touch)? | |
| Nov 24, 2020 at 5:35 | history | bounty ended | Benoît Kloeckner | ||
| Nov 20, 2020 at 18:53 | comment | added | Louis D | I updated my answer yet again to include more detailed information about the case of larger testing capacity. | |
| Nov 20, 2020 at 18:52 | history | edited | Louis D | CC BY-SA 4.0 |
added yet more information
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| Nov 20, 2020 at 17:36 | history | edited | Louis D | CC BY-SA 4.0 |
added 6 characters in body
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| Nov 20, 2020 at 14:59 | history | edited | Louis D | CC BY-SA 4.0 |
updated my answer yet again with more information
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| Nov 20, 2020 at 12:12 | comment | added | Louis D | Another thing I just realized is that the "correct" comparison to the hypercube design with $D=3$ is $H_{8,2}$. $H_{8,2}$ is 7-uniform, 2-regular, has 28 vertices, 8 edges giving a compression ratio of 2/7. So it can test more people (28 instead of 27), with fewer tests (8 instead of 9), the size of each test is smaller (7 instead of 9), and the samples need to be split a fewer number of times (2 instead of 3). | |
| Nov 20, 2020 at 1:37 | comment | added | Louis D | I'll just make a comment until I have time to work out the details. In order to generalize this to higher detection capacity, say $c=2$, it seems that we need a generalization of Sperner's theorem which gives the maximum number of edges in a hypergraph $H$ such that for distinct $e,f\in E(H)$ and (not-necessarily-distinct) $g_1,g_2\in E(H)$, $e\cup g_1\not\subseteq f\cup g_2$. The dual of such a hypergraph would have a detection capacity of at least 2. As far as I can tell, the dual of the complete $k$-regular hypergraph on $2k$ vertices has a detection capacity of 1. | |
| Nov 19, 2020 at 18:10 | comment | added | Benoît Kloeckner | Good! Corollary is more straightforward than that, since edges of $G$ are exactly the $x^*$ where $x$ runs over vertices of $H$. It seems that the duals of the complete $k$-regular hypergraphs on $2k$ vertices should be investigated; they achieve best compression rate given their order, and I wonder what is their detection capacity. | |
| Nov 19, 2020 at 16:38 | comment | added | Louis D | I updated my answer. Now I want to think about the case of larger values of c. | |
| Nov 19, 2020 at 16:35 | history | edited | Louis D | CC BY-SA 4.0 |
Updated answer with new information
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| Nov 19, 2020 at 8:32 | comment | added | Benoît Kloeckner | It would be nice to have an idea of the asymptotic behavior of $c$ when $k$ is fixed and $n\to\infty$. Indeed, the bound $c\ge 1$ only gives $H(c/v)=\Omega(\log(n)/n^k)$ while $r=O(1/n^{k-1})$, so we are off by a factor $O(n/\log(n))$. | |
| Nov 19, 2020 at 8:25 | comment | added | Benoît Kloeckner | Thanks, this is very nice! The compression rate has to be compared with $H(c/v)$, but assuming $c=1$ in the above example, $H_{6,3}$ is only a factor $\simeq 1.05$ off and $H_{7,3}$ only $\simeq 1.07$ compared to a factor $\simeq1.46$ for the hypercube of dimension $3$, so these examples are very good! (They have $c=1$, otherwise they would probably beat the lower bound.) | |
| Nov 19, 2020 at 3:12 | history | answered | Louis D | CC BY-SA 4.0 |