Catherine S. Greenhill

On Markov Chains for Independent Sets

Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby?Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing time of the new chain for a certain range of values of the parameter ?. This range is wider than the range for which the mixing time of the Luby?Vigoda chain is known to be polynomially bounded. Moreover, the upper bound on the mixing time of the new chain is always smaller than the best known upper bound on the mixing time of the Luby?Vigoda chain for larger values of ? (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov chains for independent sets, a simple insert/delete chain, and the new chain, to show that the insert/delete chain is rapidly mixing for a wider range of values of ? than was previously known.

The Relative Complexity of Approximate Counting Problems

AbstractTwo natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of efficient approximation algorithm known as an “FPRAS”, and (ii) those that are complete for #P with respect to approximation-preserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically defined subclass of #P.

Sampling Regular Graphs and a Peer-to-Peer Network

We consider a simple Markov chain for d-regular graphs on n vertices, and show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. A related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even degree d. We use this to model a certain peer-to-peer network structure. We prove that the related chain has mixing time which is bounded by a polynomial in N, the expected number of vertices, under reasonable assumptions about the arrival and departure process.

The complexity of counting colourings and independent sets in sparse graphs and hypergraphs

Abstract. We consider certain counting problems involving colourings of graphs and independent sets in hypergraphs. Using polynomial interpolation techniques, we show that these problems are #P-complete. Therefore, efficient approximate counting is the most one can realistically expect to achieve. Rapidly mixing Markov chains which can be used for approximately solving some of these counting problems have been recently developed by the author and others.

Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums

Let s = (s1,.....,sm) and t = (t1,.....,tn) be vectors of non-negative integer-valued functions with equal sum S = Σi=1m si = Σj = 1n tj. Let N(s,t) be the number of m × n matrices with entries from {0, 1} such that the ith row has row sum si and the jth column has column sum tj. Equivalently, N(s, t) is the number of labelled bipartite graphs with degrees of the vertices in one side of the bipartition given by s and the degrees of the vertices in the other side given by t. We give an asymptotic formula for N(s, t) which holds when S → ∞ with 1 ≤ st = o(S2/3), where s = maxi si and t = maxj tj. This extends a result of McKay and Wang [Linear Algebra Appl. 373 (2003) 273-288] for the semiregular case (when si = s for 1 ≤ i ≤ m and tj = t for 1 ≤ j ≤ n). The previously strongest result for the non-semiregular case required 1 ≤ max{s, t} = o(S1/4), due to McKay [Enumeration and Design, Academic Press, Canada, 1984, pp. 225-238].

Generation of simple quadrangulations of the sphere

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamotos proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batageljs proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm.

Surveys in Combinatorics, 1999: Random Walks on Combinatorial Objects

Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we re-examine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem.

An Extension of Path Coupling and Its Application to the Glauber Dynamics for Graph Colorings

A new method for analyzing the mixing time of Markov chains is described. This method is an extension of path coupling and involves analyzing the coupling over multiple steps. The expected behavior of the coupling at a certain stopping time is used to bound the expected behavior of the coupling after a fixed number of steps. The new method is applied to analyze the mixing time of the Glauber dynamics for graph colorings. We show that the Glauber dynamics has O(n log(n)) mixing time for triangle-free

Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums

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Polynomial-time counting and sampling of two-rowed contingency tables

-regular graphs if k colors are used, where