Rounds in Combinatorial Search

The search complexity of a separating system ${cal H} subseteq 2^{[m]}$ is the minimum number of questions of type ``$xin H$? hinspace \u27\u27 (where $H in {cal H}$) needed in the worst case to determine a hidden element $xin [m]$. If we are allowed to ask the questions in at most $k$ batches then we speak of the emph{$k$-round} (or emph{$k$-stage}) complexity of ${cal H}$, denoted by $hbox{c}_k ({cal H})$. While $1$-round and $m$-round complexities (called non-adaptive and adaptive complexities, respectively) are widely studied (see for example Aigner cite{A}), much less is known about other possible values of $k$, though the cases with small values of $k$ (tipically $k=2$) attracted significant attention recently, due to their applications in DNA library screening. It is clear that $|{cal H}| geq hbox{c}_{1} ({cal H}) geq hbox{c}_{2} ({cal H}) geq ldots geq hbox{c}_{m} ({cal H})$. A group of problems raised by {G. O. H. Katona} cite{Ka} is to characterize those separating systems for which some of these inequalities are tight. In this paper we are discussing set systems ${cal H}$ with the property $|{cal H}| = hbox{c}_{k} ({cal H})$ for any $kgeq 3$. We give a necessary condition for this property by proving a theorem about traces of hypergraphs which also has its own interest

Separation with restricted families of sets

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one can select $\lceil\log n\rceil+1$ members of $\mathcal F$ that separate $X$. If $|\mathcal F|\ge \alpha 2^n$ for some $0<\alpha<1/2$, then $\log n+O(\log\frac1{\alpha}\log\log\frac1{\alpha})$ members of $\mathcal F$ are always sufficient to separate all pairs of elements of $X$ that are separated by some member of $\mathcal F$. This result is generalized to simultaneous separation in several sets. Analogous questions on separation by families of bounded Vapnik-Chervonenkis dimension and separation of point sets in ${\mathbb{R}}^d$ by convex sets are also considered.Comment: 13 page

Rounds in a combinatorial search problem

We consider the following combinatorial search problem: we are given some excellent elements of $[n]$ and we should find at least one, asking questions of the following type: "Is there an excellent element in $A \subset [n]$?". G.O.H. Katona proved sharp results for the number of questions needed to ask in the adaptive, non-adaptive and two-round versions of this problem. We verify a conjecture of Katona by proving that in the $r$-round version we need to ask $rn^{1/r}+O(1)$ queries for fixed $r$ and this is sharp. We also prove bounds for the queries needed to ask if we want to find at least $d$ excellent elements.Comment: 14 page