Improved bounds for mixing rates of Markov chains and multicommodity flow

Abstract

In recent years, Markov chain simulation has emerged as a powerful algorithmic paradigm. Its chief application is to the random sampling of combinatorial structures from a specified probability distribution. Such a sampling procedure lies at the heart of efficient probabilistic algorithms for a wide variety of problems, such as approximating the size of combinatorially defined sets, estimating the expectation of certain operators in statistical physics, and combinatorial optimisation by stochastic search.

The algorithmic idea is simple. Suppose we wish to sample the elements of a large but finite set X of structures from a distribution π. First, construct a Markov chain whose states are the elements of X and which converges asymptotically to the stationary or equilibrium distribution π over X; it is usually possible to do this using as transitions simple random perturbations of the structures in X. Then, starting from an arbitrary state, simulate the chain until it is close to equilibrium: the distribution of the final state will be close to the desired distribution π.

To take a typical example, let H be a connected graph and X the set of spanning trees of H, and suppose we wish to sample elements of X from a uniform distribution. Consider the Markov chain MC(H) with state space X which, given a spanning tree T ε X, makes transitions as follows: select uniformly at random an edge e of H which does not belong to T, add e to T, thereby creating a single cycle C, and finally remove an edge of C uniformly at random to create a new spanning tree T′. It is not hard to check that this Markov chain converges to the uniform distribution over X.

Analysing the efficiency of the above technique in a given application presents a considerable challenge. The key issue is to determine the mixing rate of the chain, i.e., the number of simulation steps needed to ensure that it is sufficiently close to its equilibrium distribution π. An efficient algorithm can result only if this number is reasonably small, which usually means dramatically less than the size of the state space X itself. For example, in the spanning tree problem above we would want MC(H) to reach equilibrium in time bounded by some polynomial in n, the size of the problem instance H; however, the number of states ¦X¦ will typically be exponential in n. Informally, we will call chains having this property rapidly mixing. (More correctly, this is a property of families of chains, such as MC(H), parameterised on problem instances.)

The first analyses of the complex Markov chains arising in the combinatorial applications mentioned above were made possible using a quantity called the conductance [20,

A useful piece of technology for obtaining lower bounds on Φ in complex examples was developed in [8, 20]. The idea is to construct a canonical path γ_xy in the graph G between each ordered pair of distinct states x and y. If the paths can be chosen in such a way that no edge is overloaded by paths, then the chain cannot contain a constriction, so Φ is not too small. (The existence of a constriction between S and X−S would imply that any choice of paths must overload the edges in the constriction.) More precisely, suppose ρ is the maximum loading of an edge by paths; then it is not hard to show (see Theorem 3) that Φ ≥ (2ρ)⁻¹, so ρ does indeed provide a bound on the mixing rate of the chain. The power of this observation lies in the fact that a good collection Γ = {γ _xy } of canonical paths can sometimes be constructed for which ρ can be bounded rather tightly.

Recently Diaconis and Stroock [6] observed that path arguments similar to that described above can lead directly to bounds on the mixing rate, independently of the conductance Φ. In this paper, we present a new bound which is a modification of that of Diaconis and Stroock. The new bound also involves the maximum loading of an edge by paths, but takes into account the lengths of the paths. A simplified form of the bound (Corollary 6) relates the mixing rate to the product ρℓ for a collection of paths Γ, where ℓ is the length of a longest path in Γ. This bound turns out to be sharper than the conductance-based bound above when the maximum path length ℓ is small compared to ρ.

In Section 3 of the paper, we illustrate the effectiveness of the new bound by obtaining significantly improved estimates for the mixing rate of several important complex Markov chains, which have been used in the design of algorithms for problems involving monomerdimer systems, matchings in graphs, the Ising model, and almost uniform generation of combinatorial structures. The factors saved in the mixing rate translate directly to the runtime of the algorithms that use the chains. These improvements apparently do not follow from the similar bound given by Diaconis and Stroock.

Finally, in Section 4, we address the problem of characterising the rapid mixing property for reversible Markov chains. It is already known that the conductance Φ characterises rapid mixing, in the sense that Φ⁻¹ measures the mixing rate up to a polynomial factor (in fact, a square). We show that a similar characterisation in terms of the path measure ρ also holds, provided ρ is generalised in a natural way. To do this we view the graph G describing the Markov chain as a flow network and consider a multicommodity flow problem in which a certain quantity of some commodity (x, y) is to be transported from x to y for all pairs x, y ε X. For a given flow, ρ may then be interpreted as the maximum total flow through any edge e as a fraction of its weight, or capacity. Minimising over all possible flows, we get a quantity which we call the resistance ρ ≡ ρ(G) of the Markov chain. Our last result states that, if a Markov chain is close to equilibrium after τ steps, then its resistance cannot exceed O(τ log [itπ ⁻¹_min ), where π _min = min _x∈X π(x). Therefore, under reasonable assumptions about the stationary distribution π, the resistance also characterises the rapid mixing property. In fact it is possible to show something a little stronger: the quantities Φ⁻¹ and ρ are equal up to a factor O(logπ ⁻¹_min ). This is actually an approximate max-flow min-cut theorem for the multicommodity now problem, and generalises a result obtained in a different context by Leighton and Rao [14].

In this Extended Abstract, we omit some details in proofs and examples. For a more complete treatment the reader is referred to the full paper [21].

The author wishes to acknowledge the support of the International Computer Science Institute at Berkeley and DIMACS Center, Rutgers University.