36,327 research outputs found
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
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
A Breezing Proof of the KMW Bound
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW)
proved a hardness result for several fundamental graph problems in the LOCAL
model: For any (randomized) algorithm, there are input graphs with nodes
and maximum degree on which (expected) communication rounds are
required to obtain polylogarithmic approximations to a minimum vertex cover,
minimum dominating set, or maximum matching. Via reduction, this hardness
extends to symmetry breaking tasks like finding maximal independent sets or
maximal matchings. Today, more than years later, there is still no proof
of this result that is easy on the reader. Setting out to change this, in this
work, we provide a fully self-contained and proof of the KMW
lower bound. The key argument is algorithmic, and it relies on an invariant
that can be readily verified from the generation rules of the lower bound
graphs.Comment: 21 pages, 6 figure
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
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