24,516 research outputs found
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
Obtaining lower bounds for NP-hard problems has for a long time been an
active area of research. Recent algebraic techniques introduced by Jonsson et
al. (SODA 2013) show that the time complexity of the parameterized SAT()
problem correlates to the lattice of strong partial clones. With this ordering
they isolated a relation such that SAT() can be solved at least as fast
as any other NP-hard SAT() problem. In this paper we extend this method
and show that such languages also exist for the max ones problem
(MaxOnes()) and the Boolean valued constraint satisfaction problem over
finite-valued constraint languages (VCSP()). With the help of these
languages we relate MaxOnes and VCSP to the exponential time hypothesis in
several different ways.Comment: This is an extended version of Relating the Time Complexity of
Optimization Problems in Light of the Exponential-Time Hypothesis, appearing
in Proceedings of the 39th International Symposium on Mathematical
Foundations of Computer Science MFCS 2014 Budapest, August 25-29, 201
Sparse Signal Recovery under Poisson Statistics
We are motivated by problems that arise in a number of applications such as
Online Marketing and explosives detection, where the observations are usually
modeled using Poisson statistics. We model each observation as a Poisson random
variable whose mean is a sparse linear superposition of known patterns. Unlike
many conventional problems observations here are not identically distributed
since they are associated with different sensing modalities. We analyze the
performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting
involves a non-linear optimization but yet is computationally tractable. We
derive fundamental sample complexity bounds for sparse recovery when the
measurements are contaminated with Poisson noise. In contrast to the
least-squares linear regression setting with Gaussian noise, we observe that in
addition to sparsity, the scale of the parameters also fundamentally impacts
sample complexity. We introduce a novel notion of Restricted Likelihood
Perturbation (RLP), to jointly account for scale and sparsity. We derive sample
complexity bounds for regularized ML estimators in terms of RLP and
further specialize these results for deterministic and random sensing matrix
designs.Comment: 13 pages, 11 figures, 2 tables, submitted to IEEE Transactions on
Signal Processin
Computational complexity of the landscape I
We study the computational complexity of the physical problem of finding
vacua of string theory which agree with data, such as the cosmological
constant, and show that such problems are typically NP hard. In particular, we
prove that in the Bousso-Polchinski model, the problem is NP complete. We
discuss the issues this raises and the possibility that, even if we were to
find compelling evidence that some vacuum of string theory describes our
universe, we might never be able to find that vacuum explicitly.
In a companion paper, we apply this point of view to the question of how
early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure
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