Suppose that independent U(0, 1) weights are assigned to the \documentclass{article}\pagestyle{empty}\begin{document}${d\choose 2}n^{2}$\end{document} edges of the complete d-partite graph with n vertices in each of the d maximal independent sets. Then the expected weight of the minimum-weight perfect d-dimensional matching is at least \documentclass{article}\pagestyle{empty}\begin{document}$\frac{3}{16}n^{1-(2/d)}$\end{document}. We describe a randomized algorithm that finds a perfect d-dimensional matching whose expected weight is at most 5d3n1−(2/d)+d15 for all d≥3 and n≥1. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 50–58, 2002

We introduce a new technique for analyzing the mixing rate of Markov chains. We use it to prove that the Glauber dynamics on 2Δ-colorings of a graph with maximum degree Δ mixes in O(n log n) time. We prove the same mixing rate for the Insert/Delete/Drag chain of Dyer and Greenhill (Random Structures Algorithms 13, 285–317 (1998)) on independent sets of graphs with maximum degree 4. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 101–115, 2001

In a series of papers culminating in [11] Rödl, Ruciński and others study the thresholds of Ramsey properties of random graphs i.e. for a given graph H, when does a random graph almost surely have the property that for every coloring of its vertices (edges) in r colors there exists a monochromatic copy of H. They prove in many cases the existence of a function p(n,H) and two constants c(H) and C(H) such that a random graph with edge probability at most cp almost surely does not have this Ramsey property, whereas when the edge probability is at least Cp it almost surely has this property. We complement their results by showing that in certain cases, the multiplicative gap between upper and lower bound can be closed: There exists a function p(n) such that for every ε, a random graph with edge probability less than (1−ε)p almost surely does not have the Ramsey property, whereas when the edge probability is at least (1+ε)p it almost surely has this property. However, this is an existence result only, since our method yields no information about the value of the function p(n). © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 1–19, 2000

Poisson approximation, random graphs, Stein's method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Stein's method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.

Random Structures & AlgorithmsVolume 16, Issue 1 p. 1-3 TEN YEARS! Michal Karoński, Michal Karoński random@math.amu.edu.pl Adam Mickiewicz University, 60-769 Poznań, PolandSearch for more papers by this authorJoel Spencer, Joel Spencer spencer@cs.nyu.edu Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USASearch for more papers by this author Michal Karoński, Michal Karoński random@math.amu.edu.pl Adam Mickiewicz University, 60-769 Poznań, PolandSearch for more papers by this authorJoel Spencer, Joel Spencer spencer@cs.nyu.edu Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USASearch for more papers by this author First published: 03 January 2000 https://doi.org/10.1002/(SICI)1098-2418(200001)16:1<1::AID-RSA1>3.0.CO;2-EAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume16, Issue1January 2000Pages 1-3 RelatedInformation

This paper studies the probability that a random tournament with specified score sequence contains a specified subgraph. The exact asymptotic value is found in the case that the scores are not too far from regular and the subgraph is not too large. An n-dimensional saddle-point method is used. As a sample application, we prove that almost all tournaments with a given score sequence (not too far from regular) have a trivial automorphism group. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 47–57, 2000

A bipartite random mapping TK, L of a finite set V=V1∪V2, |V1|=K and |V2|=L, into itself assigns independently to each i∈V1 its unique image j∈V2 with probability 1/L and to each i∈V2 its unique image j∈V1 with probability 1/K. We study the connected component structure of a random digraph G(TK, L), representing TK, L, as K→∞ and L→∞. We show that, no matter how K and L tend to infinity relative to each other, the joint distribution of the normalized order statistics for the component sizes converges in distribution to the Poisson-Dirichlet distribution on the simplex ∇={{xi}: ∑ xi≤1, xi≥xi+1≥0 for every i≥1}. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 317–342, 2000

In this paper we derive limiting distributions for the costs of partial match retrievals in K-d tries. We confine ourselves to the binary case (i.e., the alphabet, from which the letters of the keys are drawn, has cardinality 2), to dimension K=2, and to the query pattern (*, S). The probabilistic model that we assume is the Bernoulli model: keys are sequences of i.i.d. random variables, which assume the values 0 and 1 with probability p and 1−p, and are pairwise independent. In the symmetric case (p=½) our analysis reveals convergence in distribution of properly normalized costs CN, as N→∞ (N is the number of keys), to the normal distribution, as well as convergence of the moments. We also consider two models dealing with flexible queries. The first, which we call “adapted,” assumes that the decision, in which subtree of a node to continue the search, is made when we reach that node, independent of all other such decisions. The second model, which we call “predictable,” assumes that that decision is made one step in advance, telling, if the search should be continued in both left (resp. both right) subtrees of the two sons, again independent of all other such decisions. In the symmetric case the limiting distributions are the same as before, and in the asymmetric case limiting distributions are given in terms of some nonnormal distribution μp, which is the unique solution of a certain stochastic fixed point equation. Again we also establish convergence of the moments. Furthermore we derive some properties of μp from its fixed point equation, such as existence and infinite differentiability of the density, and analyticity of the moment generating function, and we also determine the support of μp. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 428–459, 2000

This is a close approximation to the content of my lecture. After a brief survey of well known properties, I present some new interpretations relating to random graphs, lattice point enumeration, and chip firing games. I then examine complexity issues and concentrate in particular, on the existence of randomized approximation schemes. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 210–228, 1999

We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the near-regular case, i.e., when all the degrees are almost equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs, which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near-regular. The proof techniques we use for the two problems are similar. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 293–308, 1999