Lajos Rónyai

Norm-Graphs

We describe several variants of the norm-graphs introduced by Kollar, Ronyai, and Szabo and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 12n5/3 edges, containing no copy of K3, 3, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K3, 3 is (1+o(1))k3. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(mt) and whose discrepancy is ?(n1/2?1/2tlogn). This settles a problem of Matou?ek.

Norm-graphs and bipartite turán numbers

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|≥ct·n2−1/t which do not contain a complete bipartite graghKt,t!+1 This establishes the exact order of magnitude of the Turán numbers ex (n, Kt,s) for any fixedt and alls≥t!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.

Computing the structure of finite algebras

In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative algebras. This theory determines, mostly in a nonconstructive way, the building blocks of these algebras. Our aim is to give polynomial time algorithms to find these building blocks, the radical and the simple direct summands of the radical-free part. The radical algorithm is based on a new, tractable characterisation of the radical. The algorithm for decomposition of semisimple algebras into simple ideals involves (and generalises) factoring polynomials over the ground field. Next, we study the problem of finding zero divisors in finite algebras. We show that thisproblem is in the same complexity class as the problem of factoring polynomials over finte fields. Applications include a polynomial time Las Vegas method to find a common invariant subspace of a set of linear transformations as well as an explicit isomorphism between a given finite simple algebra and a full matrix algebra over a finite field.

On optimal completion of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. Here we study the uniqueness problem of the best completion for two weighting methods, the Eigenvector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical examples are discussed at the end of the paper.

Polynomial time solutions of some problems of computational algebra

The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building blocks. We find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las Vegas. Some of the results generalize to fields given by oracles. Some fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.

A Combinatorial Problem on Polynomials and Rational Functions

The structure of rational functions of two real variables which take few distinct values on large (finite) Cartesian products is described. As an application, a problem of G. Purdy is solved on finite subsets of the plane which determine few distinct distances.

A novel approach for failure localization in all-optical mesh networks

Achieving fast and precise failure localization has long been a highly desired feature in all-optical mesh networks. Monitoring trail (m-trail) has been proposed as the most general monitoring structure for achieving unambiguous failure localization (UFL) of any single link failure while effectively reducing the amount of alarm signals flooding the networks. However, it is critical to come up with a fast and intelligent m-trail design approach for minimizing the number of m-trails and the total bandwidth consumed, which ubiquitously determines the length of the alarm code and bandwidth overhead for the m-trail deployment, respectively. In this paper, the m-trail design problem is investigated. To gain a deeper understanding of the problem, we first conduct a bound analysis on the minimum length of alarm code of each link required for UFL on the most sparse (i.e., ring) and dense (i.e., fully meshed) topologies. Then, a novel algorithm based on random code assignment (RCA) and random code swapping (RCS) is developed for solving the m-trail design problem. The algorithm is verified by comparison to an integer linear program (ILP) approach, and the results demonstrate its superiority in minimizing the fault management cost and bandwidth consumption while achieving significant reduction in computation time. To investigate the impact of topology diversity, extensive simulation is conducted on thousands of random network topologies with systematically increased network density.

Trie: An alternative data structure for data mining algorithms

Frequent itemset mining is one of the most important data mining fields. Most algorithms are APRIORI based, where hash-trees are used extensively to speed up the search for itemsets. In this paper, we demonstrate that a version of the trie data structure outperforms hash-trees in some data mining applications. Tries appear to offer simpler and scalable algorithms which turned out to be faster for lower support thresholds. To back up our claims, we present test results based on real-life datasets.

Planar functions over finite fields

Letp>2 be a prime. A functionf: GF(p)→GF(p) is planar if for everya∃GF(p)*, the functionf(x+a−f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp2, thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.

On a Conjecture of Kemnitz

n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from (the direct sum of d copies of ) there is a subsequence of length n whose sum is 0 in . Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that . The proof uses an algebraic approach.