Renyi Institute

Budapest, Hungary

Renyi Institute

Budapest, Hungary
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Keller C.,Ben - Gurion University of the Negev | Smorodinsky S.,Ben - Gurion University of the Negev | Tardos G.,Renyi Institute
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2017

Let HDd(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(p, d + 1) is O(pd2+d). This paper has two parts. In the first part we present several improved bounds on HDd(p, q). In particular, we obtain the first near tight estimate of HDd(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem. In the second part we prove a (p, 2)-theorem for families in R2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity. Copyright © by SIAM.

Fox J.,Princeton | Pach J.,Ecole Polytechnique Federale de Lausanne | Pach J.,Renyi Institute
Combinatorics Probability and Computing | Year: 2010

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of O(m3/4√log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for anyε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 +ε )n edges. Furthermore, the number of such labelled graphs is at most (1 +ε )nT(n), where T(n) = nn-2 is the number of labelled trees on n vertices. © 20o9 Cambridge University Press.

Moser R.A.,ETH Zurich | Tardos G.,Simon Fraser University | Tardos G.,Renyi Institute
Journal of the ACM | Year: 2010

The Lovász Local Lemma discovered by Erds and Lovász in 1975 is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In 1991, József Beck was the first to demonstrate that a constructive variant can be given under certain more restrictive conditions, starting a whole line of research aimed at improving his algorithm's performance and relaxing its restrictions. In the present article, we improve upon recent findings so as to provide a method for making almost all known applications of the general Local Lemma algorithmic. © 2010 ACM.

Pach J.,Lausanne and Renyi Institute | Rubin N.,Ben - Gurion University of the Negev | Tardos G.,Renyi Institute
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2016

If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point-All other intersection points are called crossing points. The main result of this paper is a Crossing Lemma for closed curves: In any family of n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of fK(loglogn)1/8). As a corollary, we prove the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least (1 - o(l))n2.

Keszegh B.,Renyi Institute
Computational Geometry: Theory and Applications | Year: 2012

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover, we give efficient coloring algorithms. © 2012 Elsevier B.V.

Pach J.,Renyi Institute | Toth G.,Renyi Institute
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

The monotone crossing number of G is defined as the smallest number of crossing points in a drawing of G in the plane, where every edge is represented by an x-monotone curve, that is, by a connected continuous arc with the property that every vertical line intersects it in at most one point. It is shown that this parameter can be strictly larger than the classical crossing number cr(G), but it is bounded from above by 2cr 2(G). This is in sharp contrast with the behavior of the rectilinear crossing number, which cannot be bounded from above by any function of cr(G). © 2012 Springer-Verlag Berlin Heidelberg.

Katona G.O.H.,Renyi Institute
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

A model of random databases is given, with arbitrary correlations among the data of one individual. This is given by a joint distribution function. The individuals are chosen independently, their number m is considered to be (approximately) known. The probability of the event that a given functional dependency A → b holds (A is a set of attributes, b is an attribute) is determined in a limiting sense. This probability is small if m is much larger than and is large if m is much smaller than 2 H 2(A→b)/ 2 where H 2(A→b) is an entropy like functional of the probability distribution of the data. © 2012 Springer-Verlag Berlin Heidelberg.

Miklos I.,Renyi Institute | Miklos I.,Hungarian Academy of Sciences | Tannier E.,French Institute for Research in Computer Science and Automation
Bioinformatics | Year: 2010

Motivation: When comparing the organization of two genomes, it is important not to draw conclusions on their modes of evolution from a single most parsimonious scenario explaining their differences. Better estimations can be obtained by sampling many different genomic rearrangement scenarios. For this problem, the Double Cut and Join (DCJ) model, while less relevant, is computationally easier than the Hannenhalli-Pevzner (HP) model. Indeed, in some special cases, the total number of DCJ sorting scenarios can be analytically calculated, and uniformly distributed random DCJ scenarios can be drawn in polynomial running time, while the complexity of counting the number of HP scenarios and sampling from the uniform distribution of their space is unknown, and conjectured to be #P-complete. Statistical methods, like Markov chain Monte Carlo (MCMC) for sampling from the uniform distribution of the most parsimonious or the Bayesian distribution of all possible HP scenarios are required. Results: We use the computational facilities of the DCJ model to draw a sampling of HP scenarios. It is based on a parallel MCMC method that cools down DCJ scenarios to HP scenarios. We introduce two theorems underlying the theoretical mixing properties of this parallel MCMC method. The method was tested on yeast and mammalian genomic data, and allowed us to provide estimates of the different modes of evolution in diverse lineages. © The Author 2010. Published by Oxford University Press. All rights reserved.

Miklos I.,Renyi Institute | Zadori Z.,Hungarian Academy of Sciences
PLoS Computational Biology | Year: 2012

HD amino acid duplex has been found in the active center of many different enzymes. The dyad plays remarkably different roles in their catalytic processes that usually involve metal coordination. An HD motif is positioned directly on the amyloid beta fragment (Aβ) and on the carboxy-terminal region of the extracellular domain (CAED) of the human amyloid precursor protein (APP) and a taxonomically well defined group of APP orthologues (APPOs). In human Aβ HD is part of a presumed, RGD-like integrin-binding motif RHD; however, neither RHD nor RXD demonstrates reasonable conservation in APPOs. The sequences of CAEDs and the position of the HD are not particularly conserved either, yet we show with a novel statistical method using evolutionary modeling that the presence of HD on CAEDs cannot be the result of neutral evolutionary forces (p&0.0001). The motif is positively selected along the evolutionary process in the majority of APPOs, despite the fact that HD motif is underrepresented in the proteomes of all species of the animal kingdom. Position migration can be explained by high probability occurrence of multiple copies of HD on intermediate sequences, from which only one is kept by selective evolutionary forces, in a similar way as in the case of the "transcription binding site turnover." CAED of all APP orthologues and homologues are predicted to bind metal ions including Amyloid-like protein 1 (APLP1) and Amyloid-like protein 2 (APLP2). Our results suggest that HDs on the CAEDs are most probably key components of metal-binding domains, which facilitate and/or regulate inter- or intra-molecular interactions in a metal ion-dependent or metal ion concentration-dependent manner. The involvement of naturally occurring mutations of HD (Tottori (D7N) and English (H6R) mutations) in early onset Alzheimer's disease gives additional support to our finding that HD has an evolutionary preserved function on APPOs. © 2012 Miklós, Zadori.

Di Battista G.,Third University of Rome | Frati F.,University of Sydney | Pach J.,Ecole Polytechnique Federale de Lausanne | Pach J.,Renyi Institute
SIAM Journal on Computing | Year: 2013

We prove that planar graphs have O(log2 n) queue number, thus improving upon the previous O(Formula Presented) upper bound. Consequently, planar graphs admit three-dimensional straight-line crossing-free grid drawings in O(n log8 n) volume, thus improving upon the previous O(n3/2) upper bound. © 2013 Society for Industrial and Applied Mathematics.

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