Claudia Bertram-Kretzberg

The Algorithmic Aspects of Uncrowded Hypergraphs

We consider the problem of finding deterministically a large independent set of guaranteed size in a hypergraph on n vertices and with m edges. With respect to the Turan bound, the quality of our solutions is better for hypergraphs with not too many small cycles by a logarithmic factor in the input size. The algorithms are fast; they often have a running time of O(m) + o(n3). Indeed, the denser the hypergraphs are the closer the running times are to the linear times. For the first time, this gives for some combinatorial problems algorithmic solutions with state-of-the-art quality, solutions of which only the existence was known to date. In some cases, the corresponding upper bounds match the lower bounds up to constant factors. The involved concepts are uncrowded hypergraphs.

The algorithmic aspects of uncrowded hypergraphs

We consider the problem of finding deterministically a large independent set of guaranteed size in a hypergraph on n vertices and with m edges. With respect to the Turin bound, the quality of our solutions is for hypergraphs with not too many small cycles by a logarithmic factor in the input size better. The algorithms are fast. Namely, they often have a running time of O(m) + o(n{sup 3}). Indeed, the denser the hypergraphs the more close are the running times to the linear ones. This gives for the first time for some combinatorial problems algorithmic solutions with state-of-the-art quality, solutions of which so far only the existence was known. In some cases, the corresponding upper bounds match the lower bounds up to constant factors. The involved concepts are uncrowded hypergraphs.

An Algorithm for Heilbronn's Problem

Heilbronn conjectured that given arbitrary n points from R2, located in the unit square (or disc), there must be three points which form a triangle of area at most O(1/n2). This conjecture was shown to be false by a nonconstructive argument of Komlos, Pintz and Szemeredi [6] who showed that for every n there is a configuration of n points in the unit square where all triangles have area at least Ω(log n/n2). In this paper, we provide a polynomial-time algorithm which for every n computes such a configuration of n points.

Sparse 0−1 Matrices and Forbidden Hypergraphs

We consider the problem of determining the maximum number N ( m , k , r ) of columns of a 0−1 matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over ℤ 2 . For fixed integers k [ges ]4 and r [ges ]2, where k is even and gcd( k −1, r ) = 1, we prove the lower bound N ( m , k , r ) = Ω( m kr /2( k −1) ·(ln m ) 1/ k −1 ). This improves on earlier results from [14] by a factor Θ((ln m ) 1/ k −1 ). Moreover, we describe a polynomial time algorithm achieving this new lower bound.

Sparse 0-1-matrices and forbidden hypergraphs

We consider the problem of determining the maximum number N(m, k, r) of columns of a O-l-matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over Zs. For fixed integers k>4 and r>2 where k is even and gcd(k-1, r) = 1, we prove the probabilistic lower bound N(m, k, r) = R(m* . (In m) A). This improves on earlier results from [13] by the factor O((lnm)k--l). Moreover, we describe a polynomial time algorithm achieving this new lower bound.

Proper Bounded Edge-Colorings

For an n-element set X and a proper coloring ∆: [X] −→ {0, 1, . . .} where each color class is a matching with cardinality bounded by u, we show that there exists a totally multicolored subset Y ⊆ X with |Y | ≥ max { c1 · ( n/u ) 1 2k−1 , c2 · ( n/u ) 1 2k−1 · ( ln ( u/ √ n )) 1 2k−1 } This bound is tight up to constant factors for u = ω(n ) for any > 0. Moreover, for fixed k, we give a polynomial time algorithm for finding such a set Y of guaranteed size.

MOD p -tests, almost independence and small probability spaces

We consider approximations of probability distributions over ℤ p n . We present an approach to estimate the quality of approximations towards the construction of small probability spaces which are used to derandomize algorithms. In contrast to results by Even et al. [13], our methods are simple, and for reasonably small p, we get smaller sample spaces. Our considerations are motivated by a problem which was mentioned in recent work of Azar et al. [5], namely, how to construct in time polynomial in n a good approximation to the joint probability distribution of i.i.d. random variables X1,...,X n where each X i has values in {0,1}. Our considerations improve on results in [5].

Multiple Product Modulo Arbitrary Numbers

Let n binary numbers of length n be given. The Boolean function ”Multiple Product” MP n asks for (some binary representation of) the value of their product. It has been shown in [SR],[SBKH] that this function can be computed in polynomial-size threshold circuits of depth 4. For a lot of other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits.