# co.combinatorics – What are environment friendly pooling designs for RT-PCR assessments? Answer

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co.combinatorics – What are environment friendly pooling designs for RT-PCR assessments?

This is not a complete respond, however too lengthy for a observation. I suppose it comes closest to attempting to respond Question 3 or the common query of whether or not the hypercube project can breathe improved.

Definition Given a hypergraph $$G=({v_1, dots, v_n}, E)$$, the twin of $$G$$ is the hypergraph $$H$$ with $$V(H)=E(G)$$ and $$E(H)={{ein E(G): v_iin e}: iin [k]}$$ (in different phrases, every verge of $$H$$ is a maximal assortment of edges from $$G$$ that are incident with a sole vertex).

Let $$H_{n,ok}$$ breathe the twin of $$K_n^{ok}$$, the whole $$ok$$-regular hypergraph on $$n$$ vertices. Note that the twin of $$H_{n,ok}$$ is isomorphic to $$K_n^ok$$.

(It appears to me that this hypergraph will need to have been studied earlier than, however I could not discover any reference to it. One workable lead is that $$H_{4,2}$$ is what you convene the full quadrilateral.)

Claim 1.
$$H_{n,ok}$$ is a $$binom{n-1}{k-1}$$-uniform $$ok$$-regular hypergraph with $$binom{n}{ok}$$ vertices and $$n$$ edges.

Proof. In $$K_n^ok$$, each vertex is incident with $$binom{n-1}{k-1}$$ edges, each verge has organize $$ok$$, there are $$binom{n}{ok}$$ edges, and $$n$$ vertices.$$sq.$$

Claim 2. $$H_{n,ok}$$ is a pooling project.

Proof.
Every vertex in $$H_{n,ok}$$ is incident with $$ok$$ edges, so $$|x^*|=ok$$. If $$X$$ is a clique of vertices with $$|X|>1$$ (which corresponds to a clique of a couple of verge in $$K_n^ok$$, which spans greater than $$ok$$ vertices in $$K_n^ok$$) then $$|X^*|>ok$$. So $$x^*neq X^*$$ if $$|X|>1$$.$$sq.$$

The compression price of $$H_{n,ok}$$ is $$frac{n}{binom{n}{ok}}$$ which is minimized when $$ok=lfloor{n/2}rfloor$$. Also point to that ratio of the uniformity to the variety of vertices is $$binom{n-1}{k-1}/binom{n}{ok}=ok/n$$. So there’s a tradeoff
when minimizing the compression price, because the uniformity and diploma expand after we expand $$ok$$.

Some extra examples: $$H_{5,2}$$ is 4-uniform with 10 vertices and 5 edges giving a compression ratio of $$1/2$$. $$H_{6,3}$$ is 10-uniform with 20 vertices and 6 edges, giving a compression ratio of $$3/10$$. $$H_{7,3}$$ is 15-uniform with 35 vertices and seven edges, giving a compression ratio of $$1/5$$. Note that the hypercube project with $$D=3$$ is 9-regular with 27 vertices and 9 edges and thus a compression ratio of 1/3, so $$H_{6,3}$$ and $$H_{7,3}$$ examine favorably on this illustration.

Update 1. (It appears greatest to replace my earlier respond reasonably than write a fresh one.)

After considering it over some extra, I cerebrate I’ve an alternate characterization of pooling designs which each makes it simpler to bridle if $$H$$ is a pooling project and elucidates some options of pooling designs. In specific, this provides a unostentatious proof of the Propositions in your respond.

Claim 3 $$H$$ is a pooling project if and provided that $$x^*notsubseteq y^*$$ for all discrete $$x,yin V(H)$$.

Proof.
($$Rightarrow$$) Suppose there exists discrete $$x,yin V(H)$$ such that $$x^*subseteq y^*$$. Then $$y^*={x,y}^*$$ and thus $$H$$ shouldn’t be a pooling project.

($$Leftarrow$$) Suppose $$H$$ shouldn’t be a pooling project; that’s, suppose there exists $$yin V(H)$$ and $$Ysubseteq V(H)$$ with $$Yneq {y}$$ such that $$y^*=Y^*$$. Since $$Yneq {y}$$, there exists $$xin Y$$ such that $$xneq y$$. Since $$xin Y$$, we now have $$x^*subseteq Y^*=y^*$$.
$$sq.$$

Corollary 1
Let $$H$$ breathe a hypergraph and let $$G$$ breathe the twin of $$H$$.
$$H$$ is a pooling project if and provided that $$enotsubseteq f$$ for all discrete $$e,fin E(G)$$.

Proof.
($$Rightarrow$$) Suppose $$H$$ is a pooling project. Choose discrete $$e,fin E(G)$$ which coincide to discrete $$x, yin V(H)$$ respectively. Since $$x^*notsubseteq y^*$$, we now have $$enotsubseteq f$$.

($$Leftarrow$$) Suppose $$enotsubseteq f$$ for all discrete $$e,fin E(G)$$. Choose discrete $$x,yin V(H)$$ which coincide to discrete $$e,fin E(G)$$. Since $$enotsubseteq f$$, we now have $$x^*notsubseteq y^*$$.
$$sq.$$

Corollary 2
Let $$H$$ breathe a hypergraph with $$e$$ edges and $$n$$ vertices such that $$binom{e}{lfloor{e/2}rfloor}. Then $$H$$ shouldn’t be a pooling project.

Proof.
Let $$G$$ breathe the twin of $$H$$ and point to that $$G$$ has $$e$$ vertices and $$n$$ edges. Since $$|E(G)|=n>binom{e}{lfloor{e/2}rfloor}=binom{lfloor/2rfloor}$$, Sperner’s theorem implies that there exists discrete $$e,fin E(G)$$ such that $$esubseteq f$$. Thus $$H$$ shouldn’t be a pooling project by Corollary 1.
$$sq.$$

In specific, this proves that each pooling project on $$4leq nleq 6$$ vertices has no less than 4 edges, each pooling project on $$7leq nleq 10$$ vertices has no less than 5 edges, and many others.

Update 2.

Again, after contemplating some extra, I now cerebrate it is clearer to simply stay within the setting of the hypergraph $$G$$ and neglect about taking the twin.

For instance, let’s examine the $$K_8$$-design to the hypercube project with $$D=3$$. In the $$K_8$$-design, every verge is a specimen (there are 28), every vertex is a take a look at pooling the samples that are incident with that vertex (there are 8), every take a look at swimming pools 7 samples (because the diploma of every vertex is 7), and every specimen will breathe used twice (since $$K_8$$ is 2-uniform). As I discussed in a observation, that is higher than the $$D=3$$ hypercube project in each parameter. Also you may behold that if precisely one specimen is contaminated, say the verge $${i,j}$$, then precisely two assessments (take a look at $$i$$ and take a look at $$j$$) will come advocate constructive.

For one other instance, let’s examine the $$K_{13}$$-design to the hypercube project with $$D=4$$. The $$D=4$$ hypercube project handles 81 samples utilizing 12 assessments every of which has measurement 27 and every specimen is used 4 occasions. The $$K_{13}$$-design handles 78 samples utilizing 13 assessments, however every take a look at has measurement 12 and every specimen is barely used 2 occasions.

For a last instance, let’s examine the $$K_{9,9}$$-design (that’s, a whole bipartite graph with 9 vertices in every sever) to the $$D=4$$ hypercube project. The $$K_{9,9}$$-design handles 81 samples utilizing 18 assessments, every of which has measurement 9 and every specimen is used 2 occasions; nonetheless, this project has the extra characteristic that if three assessments come advocate constructive, then we are going to know precisely which two samples are contaminated. Neither the $$K_{13}$$-design, nor the $$D=4$$ hypercube project have that property.

Update 3

Given this alternate route of occupied with pooling designs, the detection capability of $$G$$ can breathe outlined as the most important integer $$c$$ such that no verge $$ein E(G)$$ is contained within the union of at most $$c$$ edges of $$E(G)setminus {e}$$. So if we would like a pooling project with testing capability $$c$$ which makes use of $$t$$ assessments, we would like a hypergraph on $$t$$ vertices with as many edges as workable such that no verge $$ein E(G)$$ is contained within the union of at most $$c$$ edges of $$E(G)setminus {e}$$. It seems that this drawback was studied in Erdős, Paul; Frankl, P.; Füredi, Z., Families of finite units wherein no clique is roofed by the union of (r) others, Isr. J. Math. 51, 79-89 (1985). ZBL0587.05021.

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