16 research outputs found
A polynomial time approximation scheme for computing the supremum of Gaussian processes
We give a polynomial time approximation scheme (PTAS) for computing the
supremum of a Gaussian process. That is, given a finite set of vectors
, we compute a -factor approximation
to deterministically in time . Previously, only a constant factor
deterministic polynomial time approximation algorithm was known due to the work
of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471]. This answers
an open question of Lee (2010) and Ding [Ann. Probab. 42 (2014) 464-496]. The
study of supremum of Gaussian processes is of considerable importance in
probability with applications in functional analysis, convex geometry, and in
light of the recent breakthrough work of Ding, Lee and Peres [Ann. of Math. (2)
175 (2012) 1409-1471], to random walks on finite graphs. As such our result
could be of use elsewhere. In particular, combining with the work of Ding [Ann.
Probab. 42 (2014) 464-496], our result yields a PTAS for computing the cover
time of bounded-degree graphs. Previously, such algorithms were known only for
trees. Along the way, we also give an explicit oblivious estimator for
semi-norms in Gaussian space with optimal query complexity. Our algorithm and
its analysis are elementary in nature, using two classical comparison
inequalities, Slepian's lemma and Kanter's lemma.Comment: Published in at http://dx.doi.org/10.1214/13-AAP997 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
New Bounds for Edge-Cover by Random Walk
We show that the expected time for a random walk on a (multi-)graph to
traverse all edges of , and return to its starting point, is at most
; if each edge must be traversed in both directions, the bound is .
Both bounds are tight and may be applied to graphs with arbitrary edge lengths,
with implications for Brownian motion on a finite or infinite network of total
edge-length
The evolution of the cover time
The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP
Mixing and relaxation time for Random Walk on Wreath Product Graphs
Suppose that G and H are finite, connected graphs, G regular, X is a lazy
random walk on G and Z is a reversible ergodic Markov chain on H. The
generalized lamplighter chain X* associated with X and Z is the random walk on
the wreath product H\wr G, the graph whose vertices consist of pairs (f,x)
where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H
and x is a vertex in G. In each step, X* moves from a configuration (f,x) by
updating x to y using the transition rule of X and then independently updating
both f_x and f_y according to the transition probabilities on H; f_z for z
different of x,y remains unchanged. We estimate the mixing time of X* in terms
of the parameters of H and G. Further, we show that the relaxation time of X*
is the same order as the maximal expected hitting time of G plus |G| times the
relaxation time of the chain on H.Comment: 30 pages, 1 figur
Adversarial scheduling analysis of Game-Theoretic Models of Norm Diffusion.
In (Istrate et al. SODA 2001) we advocated the investigation of robustness of results in the theory of learning in games under adversarial scheduling models. We provide evidence that such an analysis is feasible and can lead to nontrivial results by investigating, in an adversarial scheduling setting, Peyton Young's model of diffusion of norms . In particular, our main result incorporates contagion into Peyton Young's model.evolutionary games, stochastic stability, adversarial scheduling
Cover Time and Broadcast Time
We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows:
begin{itemize}
item For any graph of size and minimum degree , we have , where denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs.
item For any -regular (or almost -regular) graph it holds that . Together with our upper bound on , this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree , since then the cover time equals the broadcast time multiplied by (neglecting logarithmic factors).
item Conversely, for any we construct almost -regular graphs that satisfy . Since any regular expander satisfies , the strong relationship given above does not hold if is polynomially smaller than .
end{itemize}
Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)
Adversarial Scheduling Analysis of Game Theoretic Models of Norm Diffusion
In (Istrate, Marathe, Ravi SODA 2001) we advocated the investigation of
robustness of results in the theory of learning in games under adversarial
scheduling models. We provide evidence that such an analysis is feasible and
can lead to nontrivial results by investigating, in an adversarial scheduling
setting, Peyton Young's model of diffusion of norms. In particular, our main
result incorporates into Peyton Young's model