958 research outputs found
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect
Discriminants in the Grothendieck Ring
We consider the "limiting behavior" of *discriminants*, by which we mean
informally the locus in some parameter space of some type of object where the
objects have certain singularities. We focus on the space of partially labeled
points on a variety X, and linear systems on X. These are connected --- we use
the first to understand the second. We describe their classes in the
Grothendieck ring of varieties, as the number of points gets large, or as the
line bundle gets very positive. They stabilize in an appropriate sense, and
their stabilization is given in terms of motivic zeta values. Motivated by our
results, we conjecture that the symmetric powers of geometrically irreducible
varieties stabilize in the Grothendieck ring (in an appropriate sense). Our
results extend parallel results in both arithmetic and topology. We give a
number of reasons for considering these questions, and propose a number of new
conjectures, both arithmetic and topological.Comment: 39 pages, updated with progress by others on various conjecture
The Euclid-Mullin graph
We introduce the Euclid-Mullin graph, which encodes all instances of Euclid's
proof of the infinitude of primes. We investigate structural properties of the
graph both theoretically and numerically; in particular, we prove that it is
not a tree.Comment: 24 pages, 2 figures, to appear in Journal of Number Theor
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
Special Geometry and Mirror Symmetry for Open String Backgrounds with N=1 Supersymmetry
We review an approach for computing non-perturbative, exact superpotentials
for Type II strings compactified on Calabi-Yau manifolds, with extra fluxes and
D-branes on top. The method is based on an open string generalization of mirror
symmetry, and takes care of the relevant sphere and disk instanton
contributions. We formulate a framework based on relative (co)homology that
uniformly treats the flux and brane sectors on a similar footing. However, one
important difference is that the brane induced potentials are of much larger
functional diversity than the flux induced ones, which have a hidden N=2
structure and depend only on the bulk geometry. This introductory lecture is
meant for an audience unfamiliar with mirror symmetry.Comment: latex, 35p, 2 figs, refs added; brief comments about duality to
fourfolds added in the concluding sectio
Classical Solutions of the TEK Model and Noncommutative Instantons in Two Dimensions
The twisted Eguchi-Kawai (TEK) model provides a non-perturbative definition
of noncommutative Yang-Mills theory: the continuum limit is approached at large
by performing suitable double scaling limits, in which non-planar
contributions are no longer suppressed. We consider here the two-dimensional
case, trying to recover within this framework the exact results recently
obtained by means of Morita equivalence. We present a rather explicit
construction of classical gauge theories on noncommutative toroidal lattice for
general topological charges. After discussing the limiting procedures to
recover the theory on the noncommutative torus and on the noncommutative plane,
we focus our attention on the classical solutions of the related TEK models. We
solve the equations of motion and we find the configurations having finite
action in the relevant double scaling limits. They can be explicitly described
in terms of twist-eaters and they exactly correspond to the instanton solutions
that are seen to dominate the partition function on the noncommutative torus.
Fluxons on the noncommutative plane are recovered as well. We also discuss how
the highly non-trivial structure of the exact partition function can emerge
from a direct matrix model computation. The quantum consistency of the TEK
formulation is eventually checked by computing Wilson loops in a particular
limit.Comment: 41 pages, JHEP3. Minor corrections, references adde
An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse
We give an explicit realization of the "String Axiverse" discussed in
Arvanitaki et. al \cite{Arvanitaki:2009fg} by extending our previous results on
moduli stabilization in theory to include axions. We extend the analysis of
\cite{Arvanitaki:2009fg} to allow for high scale inflation that leads to a
moduli dominated pre-BBN Universe. We demonstrate that an axion which solves
the strong-CP problem naturally arises and that both the axion decay constants
and GUT scale can consistently be around GeV with a much
smaller fine tuning than is usually expected. Constraints on the Axiverse from
cosmological observations, namely isocurvature perturbations and tensor modes
are described. Extending work of Fox et. al \cite{Fox:2004kb}, we note that
{\it the observation of tensor modes at Planck will falsify the Axiverse
completely.} Finally we note that Axiverse models whose lightest axion has mass
of order eV and with decay constants of order GeV
require no (anthropic) fine-tuning, though standard unification at
GeV is difficult to accommodate.Comment: 16 pages, 8 figures, v2 References adde
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