64 research outputs found
Faster generation of random spanning trees
In this paper, we set forth a new algorithm for generating approximately
uniformly random spanning trees in undirected graphs. We show how to sample
from a distribution that is within a multiplicative of uniform in
expected time \TO(m\sqrt{n}\log 1/\delta). This improves the sparse graph
case of the best previously known worst-case bound of , which has stood for twenty years.
To achieve this goal, we exploit the connection between random walks on
graphs and electrical networks, and we use this to introduce a new approach to
the problem that integrates discrete random walk-based techniques with
continuous linear algebraic methods. We believe that our use of electrical
networks and sparse linear system solvers in conjunction with random walks and
combinatorial partitioning techniques is a useful paradigm that will find
further applications in algorithmic graph theory
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
In this paper, we study the question of how efficiently a collection of
interconnected nodes can perform a global computation in the widely studied
GOSSIP model of communication. In this model, nodes do not know the global
topology of the network, and they may only initiate contact with a single
neighbor in each round. This model contrasts with the much less restrictive
LOCAL model, where a node may simultaneously communicate with all of its
neighbors in a single round. A basic question in this setting is how many
rounds of communication are required for the information dissemination problem,
in which each node has some piece of information and is required to collect all
others. In this paper, we give an algorithm that solves the information
dissemination problem in at most rounds in a network
of diameter , withno dependence on the conductance. This is at most an
additive polylogarithmic factor from the trivial lower bound of , which
applies even in the LOCAL model. In fact, we prove that something stronger is
true: any algorithm that requires rounds in the LOCAL model can be
simulated in rounds in the GOSSIP model. We thus
prove that these two models of distributed computation are essentially
equivalent
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
We introduce a new approach to computing an approximately maximum s-t flow in
a capacitated, undirected graph. This flow is computed by solving a sequence of
electrical flow problems. Each electrical flow is given by the solution of a
system of linear equations in a Laplacian matrix, and thus may be approximately
computed in nearly-linear time.
Using this approach, we develop the fastest known algorithm for computing
approximately maximum s-t flows. For a graph having n vertices and m edges, our
algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time
\tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a
(1+\epsilon)-approximately minimum s-t cut in time
\tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this
problem as well. Previously, the best dependence on m and n was achieved by the
algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute
approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and
approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3})
A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem
We prove that with high probability over the choice of a random graph
from the Erd\H{o}s-R\'enyi distribution , the -time degree
Sum-of-Squares semidefinite programming relaxation for the clique problem
will give a value of at least for some constant
. This yields a nearly tight bound on the value of this
program for any degree . Moreover we introduce a new framework
that we call \emph{pseudo-calibration} to construct Sum of Squares lower
bounds. This framework is inspired by taking a computational analog of Bayesian
probability theory. It yields a general recipe for constructing good
pseudo-distributions (i.e., dual certificates for the Sum-of-Squares
semidefinite program), and sheds further light on the ways in which this
hierarchy differs from others.Comment: 55 page
Breaking and making quantum money: toward a new quantum cryptographic protocol
Public-key quantum money is a cryptographic protocol in which a bank can
create quantum states which anyone can verify but no one except possibly the
bank can clone or forge. There are no secure public-key quantum money schemes
in the literature; as we show in this paper, the only previously published
scheme [1] is insecure. We introduce a category of quantum money protocols
which we call collision-free. For these protocols, even the bank cannot prepare
multiple identical-looking pieces of quantum money. We present a blueprint for
how such a protocol might work as well as a concrete example which we believe
may be insecure.Comment: 14 page
Sampling with Barriers: Faster Mixing via Lewis Weights
We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope
defined by inequalities in endowed with the metric defined by the
Hessian of a convex barrier function. The advantage of RHMC over Euclidean
methods such as the ball walk, hit-and-run and the Dikin walk is in its ability
to take longer steps. However, in all previous work, the mixing rate has a
linear dependence on the number of inequalities. We introduce a hybrid of the
Lewis weights barrier and the standard logarithmic barrier and prove that the
mixing rate for the corresponding RHMC is bounded by , improving on the previous best bound of (based on the log barrier). This continues the general parallels
between optimization and sampling, with the latter typically leading to new
tools and more refined analysis. To prove our main results, we have to
overcomes several challenges relating to the smoothness of Hamiltonian curves
and the self-concordance properties of the barrier. In the process, we give a
general framework for the analysis of Markov chains on Riemannian manifolds,
derive new smoothness bounds on Hamiltonian curves, a central topic of
comparison geometry, and extend self-concordance to the infinity norm, which
gives sharper bounds; these properties appear to be of independent interest
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