Analysis of Algorithms on the Cores of Random Graphs

Abstract

The k-core 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 k-core 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 k-core of a sparse random graphs, when the average degree is roughly constant. It was used recently to rederive the threshold of appearance of the k-core (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 off-line 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(log² 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 2-core 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.