Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

Partial Covering Arrays: Algorithms and Asymptotics

A covering array $\mathsf{CA}(N;t,k,v)$ is an $N\times k$ array with entries in $\{1, 2, \ldots , v\}$, for which every $N\times t$ subarray contains each $t$-tuple of $\{1, 2, \ldots , v\}^t$ among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound $\mathsf{CAN}(t,k,v)$, the minimum number $N$ of rows of a $\mathsf{CA}(N;t,k,v)$. The well known bound $\mathsf{CAN}(t,k,v)=O((t-1)v^t\log k)$ is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set $\{1, 2, \ldots , v\}^t$ need only be contained among the rows of at least $(1-\epsilon)\binom{k}{t}$ of the $N\times t$ subarrays and (2) the rows of every $N\times t$ subarray need only contain a (large) subset of $\{1, 2, \ldots , v\}^t$. In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time

The characterization of branching dependencies

AbstractA new type of dependencies in a relational database model is introduced. If b is an attribute, A is a set of attributes then it is said that b (p,q)-depends on A, in notation A (p,q)→ b, in a database r if there are no q + 1 rows in r such that they have at most p different values in A, but q + 1 different values in b. (1,1)-dependency is the classical functional dependency. Let I(A) denote the set{b: A(p,q)→ b. The set function I(A) is characterized if p=1, 1<q; p=2, 3<q; 2<p, p2−p−1<q. Implications among (p,q)-dependencies are also determined

Partial dependencies in relational databases and their realization

AbstractWeakening the functional dependencies introduced by Amstrong we get the notion of the partial dependencies defined on the relational databases. We show that the partial dependencies can be characterized by the closure operations of the poset formed by the partial functions on the attributes of the databases. On the other hand, we give necessary and sufficient conditions so that for such a closure operation one can find on the given set of attributes a database whose partial dependencies generate the given closure operation. We also investigate some questions about how to realize certain structures related to databases by a database of minimal number of rows, columns or elements