ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...

We solve an open problem concerning the mixing time of symmetric random walk on the n- dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fully-polynomial randomized approximation scheme for counting the feasible solutions of a 0-1 knapsack problem. The results extend to the case of any xed number of hyperplanes.

Abstract. The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two

Summary 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. 1

Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great

Frequent set mining is a well-known technique to summarize binary data. However, it is an open problem how difficult it is to invert the frequent set mining, i.e., how difficult it is to find a binary data set that is compatible with frequent set mining results, the frequent sets. This inverse data mining problem is related to the questions of how well privacy is preserved in the frequent sets and how well the frequent sets characterize the original data set. In this paper we analyze the computational complexity of the problem of finding a binary data set compatible with a given collection of frequent sets and show that in many cases the problem is computationally very difficult.

For many fundamental sampling problems, the best, and often the only known, approach to solving them is to take a long enough random walk on a certain Markov chain and then return the current state of the chain. Techniques to prove how long "long enough" is, i.e., the number of steps in the chain one needs to take in order to be sufficiently close to the stationary distribution of the chain, are of great importance in obtaining estimates of running times of such sampling algorithms. In this report, we survey existing techniques to bound the mixing time of Markov chains. The mixing time of a Markov chain is exactly captured by the "spectral gap" of its underlying transition matrix. The spectral gap is closely related to a geometric parameter called "conductance" which is a measure of the "edge-expansion" of the Markov chain. Conductance also captures the mixing time up to square factors. Lower bounds on conductance, which give upper bounds on the mixing time, are typically obta...

We address the following question: is the Causal Coupling method as strong as the Conductance method in showing rapid mixing of Markov Chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum-Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. An extended abstract of this work appeared in the Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, 1999.

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