Prasad Tetali

Random walks and the effective resistance of networks

In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs. Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks. We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks. The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices.

Mathematical aspects of mixing times in Markov chains

In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.

Collisions among random walks on a graph

A token located at some vertex

Approximating Min Sum Set Cover

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Analyzing Glauber dynamics by comparison of Markov chains

of a connected, undirected graph G on n vertices is said to be taking a “random walk” on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of

Information Inequalities for Joint Distributions, With Interpretations and Applications

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Simple deterministic approximation algorithms for counting matchings

. The authors consider the following problem: Suppose that two tokens are placed on G, and at each tick of the clock a certain demon decides which of them is to make the next move. The demon is trying to keep the tokens apart as long as possible. What is the expected time M before they meet?The problem arises in the study of self-stabilizing systems, a topic of recent interest in distributed computing. Since previous upper bounds for M were exponential in n, the issue was to obtain a polynomial bound. The authors use a novel potential function argument to show that in the worst case

Communication complexity and quasi randomness

M = ( \frac{4}{27} + o ( 1 ) )n^3

Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

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Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics

Abstract The input to the min sum set cover problem is a collection of n sets that jointly cover m elements. The output is a linear order on the sets, namely, in every time step from 1 to n exactly one set is chosen. For every element, this induces a first time step by which it is covered. The objective is to find a linear arrangement of the sets that minimizes the sum of these first time steps over all elements. We show that a greedy algorithm approximates min sum set cover within a ratio of 4. This result was implicit in work of Bar-Noy, Bellare, Halldorsson, Shachnai, and Tamir (1998) on chromatic sums, but we present a simpler proof. We also show that for every ε > 0, achieving an approximation ratio of 4 – ε is NP-hard. For the min sum vertex cover version of the problem (which comes up as a heuristic for speeding up solvers of semidefinite programs) we show that it can be approximated within a ratio of 2, and is NP-hard to approximate within some constant ρ > 1.