Towards random uniform sampling of bipartite graphs with given degree sequence

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of {\em semi-regular} degree sequence. The novelty of our approach lays in the construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain explicitly our main contribution and corrected a serious flaw in the cycle decompositio

Towards random uniform sampling of bipartite graphs with given degree sequence

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on n vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in n in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method

Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms

In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time

Halmazelmélet; Partíció kalkulus, Végtelen gráfok elmélete = Set Theory; Partition Calculus , Theory of Infinite Graphs

Előzetes tervünknek megfelelően a halmazelmélet alábbi területein végeztünk kutatást és értünk el számos eredményt: I. Kombinatorika II. A valósak számsosságinvariánsai és ideálelmélet III. Halmazelméleti topológia Ezek mellett Sági Gábor kiterjedt kutatást végzett a modellelmélet területén , amely eredmények kapcsolódnak a kombinatorikához is. Eredményeinket 38 közleményben publikáltuk, amelyek majdnem mind az adott terület vezető nemzetközi lapjaiban jelentel meg (5 cikket csak benyújtottunk). Számos nemzetközi konferencián is résztvettünk, és hárman közűlünk (Juhász, Sádi, Soukup) plenáris/meghívott előadók voltak számos alkalommal. | Following our research plan, we have mainly done research -- and established a number of significant results -- in several areas of set theory: I. Combinatorics II. Cardinal invariants of the continuum and ideal theory III. Set-theoretic topology In addition to these, G. Sági has done extended research in model theory that had ramifications to combinatorics. We presented our results in 38 publications, almost all of which appeared or will appear in the leading international journals of these fields (5 of these papers have been submitted but not accepted as yet). We also participated at a number of international conferences, three of us (Juhász, Sági, Soukup) as plenary and/or invited speakers at many of these