1,204 research outputs found
Submodular Stochastic Probing on Matroids
In a stochastic probing problem we are given a universe , where each
element is active independently with probability , and only a
probe of e can tell us whether it is active or not. On this universe we execute
a process that one by one probes elements --- if a probed element is active,
then we have to include it in the solution, which we gradually construct.
Throughout the process we need to obey inner constraints on the set of elements
taken into the solution, and outer constraints on the set of all probed
elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13),
and provides a unified view of a number of problems. Thus far, all the results
falling under this general framework pertain mainly to the case in which we are
maximizing a linear objective function of the successfully probed elements. In
this paper we generalize the stochastic probing problem by considering a
monotone submodular objective function. We give a -approximation algorithm for the case in which we are given
matroids as inner constraints and matroids as outer constraints.
Additionally, we obtain an improved -approximation
algorithm for linear objective functions
A -approximation algorithm for monotone submodular maximization over a -exchange system
We consider the problem of maximizing a monotone submodular function in a
-exchange system. These systems, introduced by Feldman et al., generalize
the matroid k-parity problem in a wide class of matroids and capture many other
combinatorial optimization problems. Feldman et al. show that a simple
non-oblivious local search algorithm attains a approximation ratio
for the problem of linear maximization in a -exchange system. Here, we
extend this approach to the case of monotone submodular objective functions. We
give a deterministic, non-oblivious local search algorithm that attains an
approximation ratio of for the problem of maximizing a monotone
submodular function in a -exchange system
Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems
We present three examples of delayed bifurcations for spike solutions of
reaction-diffusion systems. The delay effect results as the system passes
slowly from a stable to an unstable regime, and was previously analysed in the
context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that
the instability would not be fully realized until the system had entered well
into the unstable regime. The bifurcation is said to have been "delayed"
relative to the threshold value computed directly from a linear stability
analysis. In contrast, we analyze the delay effect in systems of PDE's. In
particular, for spike solutions of singularly perturbed generalized
Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of
delay resulting from slow passage into regimes of oscillatory and competition
instability. In the first example, for the GM model on the infinite real line,
we analyze the delay resulting from slowly tuning a control parameter through a
Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a
finite one-dimensional domain. In this scenario, as opposed to the extrinsic
tuning of a system parameter through a bifurcation value, we analyze the delay
of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the
third example, we consider competition instabilities of the GS model triggered
by the extrinsic tuning of a feed rate parameter. In all cases, we find that
the system must pass well into the unstable regime before the onset of
instability is fully observed, indicating delay. We also find that delay has an
important effect on the eventual dynamics of the system in the unstable regime.
We give analytic predictions for the magnitude of the delays as obtained
through analysis of certain explicitly solvable nonlocal eigenvalue problems.
The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen
A bi-criteria approximation algorithm for Means
We consider the classical -means clustering problem in the setting
bi-criteria approximation, in which an algoithm is allowed to output clusters, and must produce a clustering with cost at most times the
to the cost of the optimal set of clusters. We argue that this approach is
natural in many settings, for which the exact number of clusters is a priori
unknown, or unimportant up to a constant factor. We give new bi-criteria
approximation algorithms, based on linear programming and local search,
respectively, which attain a guarantee depending on the number
of clusters that may be opened. Our gurantee is
always at most and improves rapidly with (for example:
, and ). Moreover, our algorithms have only
polynomial dependence on the dimension of the input data, and so are applicable
in high-dimensional settings
Ideology and Argument: Mitt Romney and the GOP in the 2012 Election
This paper examines four rhetorical events during the 2012 presidential election to examine how ideology and argument are informed by one another in discussions of policy during campaigns. Using the first two Republican primary debates as well as the party platform and the nomination acceptance speech by Romney, this study offers a descriptive analysis and ideological study of the arguments made in 2012 by Mitt Romney during his campaign for President. The study argues that ideological constraints prevented Mitt Romney from adequately developing substantive or flexible policy arguments during the course of the election. The study also offers a way of examining ideology from the arguments provided by candidates and party officials. This study examines the way in which taxation, government, and constitution operate to organize arguments around the relevant ideological markers. Using the Affordable Care Act as an example of these processes, this thesis will provide an explanation for the failure of purity demands and ideological argument in the context of a moderate and independent electorate
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