A universal bound for a covering in regular posets and its application to pool testing Original Research Article

Let V(n) be the set of all 2n subsets of the set Nn={1,2,…,n} and Vi(n)={x∈V(n): |x|=i}. For a fixed i=1,…,n, consider a covering operator F:Vi(n)→V(n) such that x⊆F(x) for any x∈Vi(n). Let C={F(x): x∈Vi(n)}. For any 1⩽T⩽(ni), consider the decreasing continuous function gi(T)=k+((k+1)/i)(1−α) where k and α are uniquely defined by the conditions T(ki)=α(ni), k∈{i,…,n}, and 1−i/(k+1)<α⩽1. Using averaging and linear programing it is proved that1ni ∑x∈Vi(n) |F(x)|⩾gi(|C|)⩾n|C|iwith the first inequality as an equality if and only if C is a Steiner S(i,{k,k+1},n) design with a specified distance distribution. A generalization of this result to the case of monotone left-regular n-posets is given. One of motivating applications is the problem of reconstructing an unknown binary vector x of length n using pool testing under the condition that the vectors x are distributed with probabilities p|x|(1−p)n−|x| where x∈V(n) denotes the indices of the ones (active items) in x. The bound above implies that the expected number of items which remain unresolved after application in parallel of arbitrary r pools is not less thann ∑i=1n nipi(1−p)n−i2−(r/i)−np.This improves upon an information theoretic bound for the minimum average number E(n,p) of tests to reconstruct an unknown x using two-stage pool testing and allows determination of the asymptotic behavior of E(n,p) up to a positive constant factor as n→∞ under some restrictions upon p=p(n).