27 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
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})
Randomized accuracy-aware program transformations for efficient approximate computations
Despite the fact that approximate computations have come to dominate many areas of computer science, the field of program transformations has focused almost exclusively on traditional semantics-preserving transformations that do not attempt to exploit the opportunity, available in many computations, to acceptably trade off accuracy for benefits such as increased performance and reduced resource consumption.
We present a model of computation for approximate computations and an algorithm for optimizing these computations. The algorithm works with two classes of transformations: substitution transformations (which select one of a number of available implementations for a given function, with each implementation offering a different combination of accuracy and resource consumption) and sampling transformations (which randomly discard some of the inputs to a given reduction). The algorithm produces a (1+ε) randomized approximation to the optimal randomized computation (which minimizes resource consumption subject to a probabilistic accuracy specification in the form of a maximum expected error or maximum error variance).National Science Foundation (U.S.). (Grant number CCF-0811397)National Science Foundation (U.S.). (Grant number CCF-0905244)National Science Foundation (U.S.). (Grant number CCF-0843915)National Science Foundation (U.S.). (Grant number CCF-1036241)National Science Foundation (U.S.). (Grant number IIS-0835652)United States. Dept. of Energy. (Grant Number DE-SC0005288)Alfred P. Sloan Foundation. Fellowshi
Hypercontractivity, Sum-of-Squares Proofs, and their Applications
We study the computational complexity of approximating the 2->q norm of
linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as
connections between this question and issues arising in quantum information
theory and the study of Khot's Unique Games Conjecture (UGC). We show the
following:
1. For any constant even integer q>=4, a graph is a "small-set expander"
if and only if the projector into the span of the top eigenvectors of G's
adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to
the 2->q norm will refute the Small-Set Expansion Conjecture--a close variant
of the UGC. We also show that such a good approximation can be obtained in
exp(n^(2/q)) time, thus obtaining a different proof of the known subexponential
algorithm for Small Set Expansion.
2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy
certify an upper bound on the 2->4 norm of the projector to low-degree
polynomials over the Boolean cube, as well certify the unsatisfiability of the
"noisy cube" and "short code" based instances of Unique Games considered by
prior works. This improves on the previous upper bound of exp(poly log n)
rounds (for the "short code"), as well as separates the "Sum of
Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to
require omega(1) rounds.
3. We show reductions between computing the 2->4 norm and computing the
injective tensor norm of a tensor, a problem with connections to quantum
information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to
approximate to precision inverse-polynomial in the dimension, (ii) the 2->4
norm does not have a good approximation (in the sense above) unless 3-SAT can
be solved in time exp(sqrt(n) polylog(n)), and (iii) known algorithms for the
quantum separability problem imply a non-trivial additive approximation for the
2->4 norm.Comment: v1: 52 pages. v2: 53 pages, fixed small bugs in proofs of section 6
(on UG integrality gaps) and section 7 (on 2->4 norm of random matrices).
Added comments about real-vs-complex random matrices and about the
k-extendable vs k-extendable & PPT hierarchies. v3: fixed mistakes in random
matrix section. The result now holds only for matrices with random entries
instead of random column
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Large-scale identification of genetic design strategies using local search
In the past decade, computational methods have been shown to be well suited to unraveling the complex web of metabolic reactions in biological systems. Methods based on flux–balance analysis (FBA) and bi-level optimization have been used to great effect in aiding metabolic engineering. These methods predict the result of genetic manipulations and allow for the best set of manipulations to be found computationally. Bi-level FBA is, however, limited in applicability because the required computational time and resources scale poorly as the size of the metabolic system and the number of genetic manipulations increase. To overcome these limitations, we have developed Genetic Design through Local Search (GDLS), a scalable, heuristic, algorithmic method that employs an approach based on local search with multiple search paths, which results in effective, low-complexity search of the space of genetic manipulations. Thus, GDLS is able to find genetic designs with greater in silico production of desired metabolites than can feasibly be found using a globally optimal search and performs favorably in comparison with heuristic searches based on evolutionary algorithms and simulated annealing
Large-scale identification of genetic design strategies using local search
In the past decade, computational methods have been shown to be well suited to unraveling the complex web of metabolic reactions in biological systems. Methods based on flux–balance analysis (FBA) and bi-level optimization have been used to great effect in aiding metabolic engineering. These methods predict the result of genetic manipulations and allow for the best set of manipulations to be found computationally. Bi-level FBA is, however, limited in applicability because the required computational time and resources scale poorly as the size of the metabolic system and the number of genetic manipulations increase. To overcome these limitations, we have developed Genetic Design through Local Search (GDLS), a scalable, heuristic, algorithmic method that employs an approach based on local search with multiple search paths, which results in effective, low-complexity search of the space of genetic manipulations. Thus, GDLS is able to find genetic designs with greater in silico production of desired metabolites than can feasibly be found using a globally optimal search and performs favorably in comparison with heuristic searches based on evolutionary algorithms and simulated annealing.Hertz Foundatio
Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus
Abstract. In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [13] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O ( p g/n) and vertex bisectors of size O ( √ gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. 1. Introduction. Spectra