30 research outputs found
Union-closed families of sets
Abstract3e use a lower bound on the number of small sets in an ideal to show that for each union-closed family of n sets there exists an element which belongs to at least of them, provided n is large enough
Lower bounds for identifying subset members with subset queries
An instance of a group testing problem is a set of objects \cO and an
unknown subset of \cO. The task is to determine by using queries of
the type ``does intersect '', where is a subset of \cO. This
problem occurs in areas such as fault detection, multiaccess communications,
optimal search, blood testing and chromosome mapping. Consider the two stage
algorithm for solving a group testing problem. In the first stage a
predetermined set of queries are asked in parallel and in the second stage,
is determined by testing individual objects. Let n=\cardof{\cO}. Suppose that
is generated by independently adding each x\in \cO to with
probability . Let () be the number of queries asked in the
first (second) stage of this algorithm. We show that if
, then \Exp(q_2) = n^{1-o(1)}, while there
exist algorithms with and \Exp(q_2) =
o(1). The proof involves a relaxation technique which can be used with
arbitrary distributions. The best previously known bound is q_1+\Exp(q_2) =
\Omega(p\log(n)). For general group testing algorithms, our results imply that
if the average number of queries over the course of ()
independent experiments is , then with high probability
non-singleton subsets are queried. This
settles a conjecture of Bill Bruno and David Torney and has important
consequences for the use of group testing in screening DNA libraries and other
applications where it is more cost effective to use non-adaptive algorithms
and/or too expensive to prepare a subset for its first test.Comment: 9 page
A constant lower bound for the union-closed sets conjecture
We show that for any union-closed family , there exists an which is contained
in a fraction of the sets in . This is the first known
constant lower bound, and improves upon the
bounds of Knill and W\'{o}jick. Our result
follows from an information theoretic strengthening of the conjecture.
Specifically, we show that if are independent samples from a
distribution over subsets of such that for all
and , then .Comment: 9 pages, 1 figure. (Update 11/28/22: Typos fixed, and added reference
to follow up work improving the bound and refuting Conjecture 1.
An improved lower bound for the union-closed set conjecture
Gilmer has recently shown that in any nonempty union-closed family of subsets of a finite set, there exists an element contained in at least a
proportion of the sets of . We improve the proportion from
to in this result. An improvement to
would be the Frankl union-closed set conjecture. We follow
Gilmer's method, replacing one key estimate by a sharp estimate. We then
suggest a new addition to this method and sketch a proof that it can obtain a
constant strictly greater than . We also disprove a
conjecture of Gilmer that would have implied the union-closed set conjecture.Comment: 10 page