Attila Sali

Design type problems motivated by database theory

Abstract Let k⩽n, p⩽q • for any choice of k distinct columns c1,c2,…,ck, there are q+1 rows such that the number of different entries in c i (1⩽i⩽k−1) in these rows is at most p, while all q+1 entries of ck in these rows are different; • this is true for no choice of k+1 distinct columns. We review results minimizing m, given n,p,q,k. Two of the results are new. The optimal or nearly optimal constructions can be considered as n partitions of the m-element set satisfying certain conditions. This version leads to the orthogonal double covers, also surveyed here.

The characterization of branching dependencies

Abstract A 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

Small Forbidden Configurations

In the present paper we continue the work begun by Sauer, Perles, Shelah and Anstee on forbidden configurations of 0–1 matrices. We give asymptotically exact bounds for all possible 2 × l forbidden submatrices and almost all 3 × l ones. These bounds are improvements of the general bounds, or else new constructions show that the general bound is best possible. It is interesting to note that up to the present state of our knowledge every forbidden configuration results in polynomial asymptotic.

Orientations of Self-complementary Graphs and the Relation of Sperner and Shannon Capacities

We prove that the edges of a self-complementary graph and its complement can be oriented in such a way that they remain isomorphic as digraphs and their union is a transitive tournament. This result is used to explore the relation between the Shannon and Sperner capacity of certain graphs. In particular, using results of Lovasz, we show that the maximum Sperner capacity over all orientations of the edges of a vertex-transitive self-complementary graph equals its Shannon capacity.

Small Forbidden Configurations IV: The 3 Rowed Case

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k×l (0,1)-matrix (the forbidden configuration). Small refers to the size of k and in this paper k = 3. Assume A is an m×n simple matrix which has no submatrix which is a row and column permutation of F. We define forb(m,F) as the best possible upper bound on n, for such a matrix A, which depends on m and F. We complete the classification for all 3-rowed (0,1)-matrices of forb (m,F) as either Θ(m), Θ(m2) or Θ(m3) (with constants depending on F).

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k−i-intersecting family.Furthermore, we discuss and Erdős-Ko-Rado-type question forL(A), as well.

Minimal Keys in Higher-Order Datamodels

We study keys in higher-order datamodels. We show that they are equivalent with certain ideals. Based on that we introduce an ordering between key sets, and investigate systems of minimal keys. We give a sufficient condition for a Sperner-family of SHL-ideals being system of minimal keys, and give lower and upper bounds for the size of the smallest Armstrong-instance.

Semantic Matching Strategies for Job Recruitment: A Comparison of New and Known Approaches

A profile describes a set of skills a person may have or a set of skills required for a particular job. Profile matching aims to determine how well a given profile fits to a requested profile. The research reported in this paper starts from exact matching measure of [21]. It is extended then by matching filters in ontology hierarchies, since profiles naturally determine filters in the subsumption relation. Next we take into consideration similarities between different skills that are not related by the subsumption relation. Finally, a totally different approach, probabilistic matching based on the maximum entropy model is analyzed.

On the existence of armstrong instances with bounded domains

The existence of Armstrong-instances of bounded domains is investigated for specific key systems. This leads to the concept of Armstrong(q, k, n)-codes. These are q-ary codes of length n, minimum distance n - k + 1 and have the property that for any possible k - 1 coordinate positions there are two codewords that agree exactly there. We derive upper and lower bounds on the length of the code as function of q and k. The upper bounds use geometric arguments and bounds on spherical codes, the lower bounds are probabilistic.

New type of coding problem motivated by database theory

The present paper is intended to survey the interaction between relational database theory and coding theory. In particular it is shown how an extremal problem for relational databases gives rise to a new type of coding problem. The former concerns minimal representation of branching dependencies that can be considered as a data mining type question. The extremal configurations involve d-distance sets in the space of disjoint pairs of k-element subsets of an n-element set X. Let X be an n-element finite set, 0 < k < n/2 an integer. Suppose that {A1, B1} and {A2, B2} are pairs of disjoint k-element subsets of X (that is, |A1| = |B1| = |A2| = |B2| = k, A1 ∩ B1 = 0, A2 ∩ B2 = 0). Define the distance of these pairs by d({A1, B1}, {A2, B2}) = min{|A1 - A2| + |B1 - B2|, |A1 - B2| + |B1 - A2|}.