Analysis of Algorithms on the Cores of Random Graphs
 Nick Wormald
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Abstract
The kcore of a graph is the largest subgraph of minimum degree at least k. It can be found by successively deleting all vertices of degree less than k.
The threshold of appearance of the kcore in a random graph was originally determined by Pittel, Spencer and the speaker. The original derivation used approximation of the vertex deletion process by differential equations. Many other papers have recently given alternative derivations.
A pseudograph model of random graphs introduced by Bollobás and Frieze, and also Chvátal, is useful for simplifying the original derivation. This model is especially useful for analysing algorothms on the kcore of a sparse random graphs, when the average degree is roughly constant. It was used recently to rederive the threshold of appearance of the kcore (with J. Cain). In addition, the following have recently been obtained concerning either of the random graphs G = G (n,c/n), c > 1, or G = G(n,m), m = cn /2.
(i) Analysis of a fast algorithm for offline load balancing when each item has a choice of two servers. This enabled us to determine the threshold of appearance of a subgraph with average degree at least 2k in the random graph (with P. Sanders and J. Cain),
(ii) Bounds on the mixing time for the giant component of a random graph. We show that with high probability the random graph has a subgraph H with “good” expansion properties and such that G–H has only “small” components with “not many” such components attached to any vertex of H. Amongst other things, this implies that the mixing time of the random walk on G is Θ(log^{2} n) (obtained recently and independently by Fountoulakis and Reed). This work is joint with I. Benjamini and G. Kozma. The subgraph is found by successively deleting the undesired vertices from the 2core of the random graph.
(iii)Lower bounds on longest cycle lengths in the random graph. These depend on the expected average degree c and improve the existing results that apply to small c>1 (by Ajtai, Komlós and Szemerédi, Fernandez de la Vega, and Suen). The new bounds arise from analysis of random greedy algorithms. Suen’s bounds for induced cycles are also improved using similar random greedy algorithms. This is joint work with J.H. Kim.
In all cases the analysis is by use of differential equations approximating relevant random variables during the course of the algorithm. Typically, this determines the performance of the algorithms accurately, even if the best bounds are not necessarily achieved by these algorithms.
 Title
 Analysis of Algorithms on the Cores of Random Graphs
 Book Title
 Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
 Book Subtitle
 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006, Barcelona, Spain, August 2830 2006. Proceedings
 Pages
 p 2
 Copyright
 2006
 DOI
 10.1007/11830924_2
 Print ISBN
 9783540380443
 Online ISBN
 9783540380450
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 4110
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Josep Díaz ^{(16)}
 Klaus Jansen ^{(17)}
 José D. P. Rolim ^{(18)}
 Uri Zwick ^{(19)}
 Editor Affiliations

 16. Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya
 17. Institute for Computer Science, University of Kiel
 18. Centre Universitaire d’Informatique
 19. School of Computer Science, Tel Aviv University
 Authors

 Nick Wormald ^{(20)}
 Author Affiliations

 20. Dept. Combinatorics and Optimization, University of Waterloo,
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