64,365 research outputs found
The Complexity of Approximately Counting Stable Roommate Assignments
We investigate the complexity of approximately counting stable roommate
assignments in two models: (i) the -attribute model, in which the preference
lists are determined by dot products of "preference vectors" with "attribute
vectors" and (ii) the -Euclidean model, in which the preference lists are
determined by the closeness of the "positions" of the people to their
"preferred positions". Exactly counting the number of assignments is
#P-complete, since Irving and Leather demonstrated #P-completeness for the
special case of the stable marriage problem. We show that counting the number
of stable roommate assignments in the -attribute model () and the
3-Euclidean model() is interreducible, in an approximation-preserving
sense, with counting independent sets (of all sizes) (#IS) in a graph, or
counting the number of satisfying assignments of a Boolean formula (#SAT). This
means that there can be no FPRAS for any of these problems unless NP=RP. As a
consequence, we infer that there is no FPRAS for counting stable roommate
assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we
give an approximation-preserving reduction from counting the number of
independent sets in a bipartite graph (#BIS) to counting the number of stable
roommate assignments both in the 3-attribute model and in the 2-Euclidean
model. #BIS is complete with respect to approximation-preserving reductions in
the logically-defined complexity class #RH\Pi_1. Hence, our result shows that
an FPRAS for counting stable roommate assignments in the 3-attribute model
would give an FPRAS for all of #RH\Pi_1. We also show that the 1-attribute
stable roommate problem always has either one or two stable roommate
assignments, so the number of assignments can be determined exactly in
polynomial time
Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators
The linear noise approximation (LNA) offers a simple means by which one can
study intrinsic noise in monostable biochemical networks. Using simple physical
arguments, we have recently introduced the slow-scale LNA (ssLNA) which is a
reduced version of the LNA under conditions of timescale separation. In this
paper, we present the first rigorous derivation of the ssLNA using the
projection operator technique and show that the ssLNA follows uniquely from the
standard LNA under the same conditions of timescale separation as those
required for the deterministic quasi-steady state approximation. We also show
that the large molecule number limit of several common stochastic model
reduction techniques under timescale separation conditions constitutes a
special case of the ssLNA.Comment: 10 pages, 1 figure, submitted to Physical Review E; see also BMC
Systems Biology 6, 39 (2012
External fluctuations in front dynamics with inertia: The overdamped limit
We study the dynamics of fronts when both inertial effects and external
fluctuations are taken into account. Stochastic fluctuations are introduced as
multiplicative noise arising from a control parameter of the system. Contrary
to the non-inertial (overdamped) case, we find that important features of the
system, such as the velocity selection picture, are not modified by the noise.
We then compute the overdamped limit of the underdamped dynamics in a more
careful way, finding that it does not exhibit any effect of noise either. Our
result poses the question as to whether or not external noise sources can be
measured in physical systems of this kind.Comment: 4 pages, 1 figure, accepted for publication in European Physical
Journal
Decoherence from internal degrees of freedom for cluster of mesoparticles : a hierarchy of master equations
A mesoscopic evolution equation for an ensemble of mesoparticles follows
after the elimination of internal degrees of freedom. If the system is composed
of a hierarchy of scales, the reduction procedure could be worked repeatedly
and the characterization of this iterating method is carried out. Namely, a
prescription describing a discrete hierarchy of master equations for the
density operator is obtained. Decoherence follows from the irreversible
coupling of the systems, defined by mesoscopic variables, to internal degrees
of freedom. We discuss briefly the existence of systems with the same dynamics
laws at different scales. We made an explicit calculation for an ensemble of
particles with internal harmonic interaction in an external anharmonic field.
New conditions related to the semiclassical limit for mesoscopic systems
(Wigner-function) are conjectured.Comment: 19 pages, 0 figures, late
An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance
We prove an optimal lower bound on the randomized communication
complexity of the much-studied Gap-Hamming-Distance problem. As a consequence,
we obtain essentially optimal multi-pass space lower bounds in the data stream
model for a number of fundamental problems, including the estimation of
frequency moments.
The Gap-Hamming-Distance problem is a communication problem, wherein Alice
and Bob receive -bit strings and , respectively. They are promised
that the Hamming distance between and is either at least
or at most , and their goal is to decide which of these is the
case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS,
2003), it had been conjectured that the naive protocol, which uses bits of
communication, is asymptotically optimal. The conjecture was shown to be true
in several special cases, e.g., when the communication is deterministic, or
when the number of rounds of communication is limited.
The proof of our aforementioned result, which settles this conjecture fully,
is based on a new geometric statement regarding correlations in Gaussian space,
related to a result of C. Borell (1985). To prove this geometric statement, we
show that random projections of not-too-small sets in Gaussian space are close
to a mixture of translated normal variables
Covariate dimension reduction for survival data via the Gaussian process latent variable model
The analysis of high dimensional survival data is challenging, primarily due
to the problem of overfitting which occurs when spurious relationships are
inferred from data that subsequently fail to exist in test data. Here we
propose a novel method of extracting a low dimensional representation of
covariates in survival data by combining the popular Gaussian Process Latent
Variable Model (GPLVM) with a Weibull Proportional Hazards Model (WPHM). The
combined model offers a flexible non-linear probabilistic method of detecting
and extracting any intrinsic low dimensional structure from high dimensional
data. By reducing the covariate dimension we aim to diminish the risk of
overfitting and increase the robustness and accuracy with which we infer
relationships between covariates and survival outcomes. In addition, we can
simultaneously combine information from multiple data sources by expressing
multiple datasets in terms of the same low dimensional space. We present
results from several simulation studies that illustrate a reduction in
overfitting and an increase in predictive performance, as well as successful
detection of intrinsic dimensionality. We provide evidence that it is
advantageous to combine dimensionality reduction with survival outcomes rather
than performing unsupervised dimensionality reduction on its own. Finally, we
use our model to analyse experimental gene expression data and detect and
extract a low dimensional representation that allows us to distinguish high and
low risk groups with superior accuracy compared to doing regression on the
original high dimensional data
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