31,452 research outputs found
A Logarithmic Bound for Solving Subset Sum with P Systems
The aim of our paper is twofold. On one hand we prove the
ability of polarizationless P systems with dissolution and with division
rules for non-elementary membranes to solve NP-complete problems in
a polynomial number of steps, and we do this by presenting a solution to
the Subset Sum problem. On the other hand, we improve some similar
results obtained for different models of P systems by reducing the number
of steps and the necessary resources to be of a logarithmic order with
respect to k (recall that n and k are the two parameters used to indicate
the size of an instance of the Subset Sum problem).
As the model we work with does not allow cooperative rules and
does not consider the membranes to have an associated polarization,
the strategy that we will follow consists on using objects to represent
the weights of the subsets through their multiplicities, and comparing the
number of objects against a fixed number of membranes. More precisely,
we will generate k membranes in log k steps.Ministerio de Educación y Ciencia TIN2006-13425Junta de Andalucía TIC-58
A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion
We consider in this paper the problem of noisy 1-bit matrix completion under
a general non-uniform sampling distribution using the max-norm as a convex
relaxation for the rank. A max-norm constrained maximum likelihood estimate is
introduced and studied. The rate of convergence for the estimate is obtained.
Information-theoretical methods are used to establish a minimax lower bound
under the general sampling model. The minimax upper and lower bounds together
yield the optimal rate of convergence for the Frobenius norm loss.
Computational algorithms and numerical performance are also discussed.Comment: 33 pages, 3 figure
Eigenmodes of the damped wave equation and small hyperbolic subsets
We study stationary solutions of the damped wave equation on a compact and
smooth Riemannian manifold without boundary. In the high frequency limit, we
prove that a sequence of -damped stationary solutions cannot be
completely concentrated in small neighborhoods of a small fixed hyperbolic
subset made of -damped trajectories of the geodesic flow. The article
also includes an appendix (by S. Nonnenmacher and the author) where we
establish the existence of an inverse logarithmic strip without eigenvalues
below the real axis, under a pressure condition on the set of undamped
trajectories.Comment: 24 pages. With an appendix by S. Nonnenmacher and the author. In this
new version, we modified an uncorrect exponent in the statement of Theorem
A.1 from the appendi
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Changing Bases: Multistage Optimization for Matroids and Matchings
This paper is motivated by the fact that many systems need to be maintained
continually while the underlying costs change over time. The challenge is to
continually maintain near-optimal solutions to the underlying optimization
problems, without creating too much churn in the solution itself. We model this
as a multistage combinatorial optimization problem where the input is a
sequence of cost functions (one for each time step); while we can change the
solution from step to step, we incur an additional cost for every such change.
We study the multistage matroid maintenance problem, where we need to maintain
a base of a matroid in each time step under the changing cost functions and
acquisition costs for adding new elements. The online version of this problem
generalizes online paging. E.g., given a graph, we need to maintain a spanning
tree at each step: we pay for the cost of the tree at time
, and also for the number of edges changed at
this step. Our main result is an -approximation, where is
the number of elements/edges and is the rank of the matroid. We also give
an approximation for the offline version of the problem. These
bounds hold when the acquisition costs are non-uniform, in which caseboth these
results are the best possible unless P=NP.
We also study the perfect matching version of the problem, where we must
maintain a perfect matching at each step under changing cost functions and
costs for adding new elements. Surprisingly, the hardness drastically
increases: for any constant , there is no
-approximation to the multistage matching maintenance
problem, even in the offline case
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