Martin E. Dyer

Randomized greedy matching. II

We consider the following randomized algorithm for finding a matching M in an arbitrary graph G = (V, E). Repeatedly, choose a random vertex u, then a random neighbour v of u. Add edge {u, v} to M and delete vertices u, v from G along with any vertices that become isolated. Our main result is that there exists a positive constant ϵ such that the expected ratio of the size of the matching produced to the size of largest matching in G is at least 0.5 + ϵ. We obtain stronger results for sparse graphs and trees and consider extensions to hypergraphs.

Approximately Counting Hamilton Paths and Cycles in Dense Graphs

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree

Randomly colouring random graphs

(\frac{1}{2}+\a)n

Computing the volume of convex bodies : a case where randomness provably helps

, for any fixed a > 0. We show that the exact counting problems are \#P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs.

Path Coupling, Dobrushin Uniqueness, and Approximate Counting

We consider the problem of generating a colouring of the random graph Gn,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are β∆ colours available, where ∆ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for Gn,p in cases where np is relatively small. Here we consider the “dense” cases, where np ∈ [ω lnn, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques.