31,081 research outputs found
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
Dust settling in local simulations of turbulent protoplanetary disks
In this paper, we study the effect of MHD turbulence on the dynamics of dust
particles in protoplanetary disks. We vary the size of the particles and relate
the dust evolution to the turbulent velocity fluctuations. We performed
numerical simulations using two Eulerian MHD codes, both based on finite
difference techniques: ZEUS--3D and NIRVANA. These were local shearing box
simulations incorporating vertical stratification. Both ideal and non ideal MHD
simulations with midplane dead zones were carried out. The codes were extended
to incorporate different models for the dust as an additional fluid component.
Good agreement between results obtained using the different approaches was
obtained. The simulations show that a thin layer of very small dust particles
is diffusively spread over the full vertical extent of the disk. We show that a
simple description obtained using the diffusion equation with a diffusion
coefficient simply expressed in terms of the velocity correlations accurately
matches the results. Dust settling starts to become apparent for particle sizes
of the order of 1 to 10 centimeters for which the gas begins to decouple in a
standard solar nebula model at 5.2 AU. However, for particles which are 10
centimeters in size, complete settling toward a very thin midplane layer is
prevented by turbulent motions within the disk, even in the presence of a
midplane dead zone of significant size. These results indicate that, when
present, MHD turbulence affects dust dynamics in protoplanetary disks. We find
that the evolution and settling of the dust can be accurately modelled using an
advection diffusion equation that incorporates vertical settling. The value of
the diffusion coefficient can be calculated from the turbulent velocity field
when that is known for a time of several local orbits.Comment: 15 pages, 16 figures, accepted in Astronomy & Astrophysic
Improving the smoothed complexity of FLIP for max cut problems
Finding locally optimal solutions for max-cut and max--cut are well-known
PLS-complete problems. An instinctive approach to finding such a locally
optimum solution is the FLIP method. Even though FLIP requires exponential time
in worst-case instances, it tends to terminate quickly in practical instances.
To explain this discrepancy, the run-time of FLIP has been studied in the
smoothed complexity framework. Etscheid and R\"{o}glin showed that the smoothed
complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel,
Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut
in complete graphs is , where is an upper bound on
the random edge-weight density and is the number of vertices in the input
graph.
While Angel et al.'s result showed the first polynomial smoothed complexity,
they also conjectured that their run-time bound is far from optimal. In this
work, we make substantial progress towards improving the run-time bound. We
prove that the smoothed complexity of FLIP in complete graphs is . Our results are based on a carefully chosen matrix whose rank
captures the run-time of the method along with improved rank bounds for this
matrix and an improved union bound based on this matrix. In addition, our
techniques provide a general framework for analyzing FLIP in the smoothed
framework. We illustrate this general framework by showing that the smoothed
complexity of FLIP for max--cut in complete graphs is polynomial and for
max--cut in arbitrary graphs is quasi-polynomial. We believe that our
techniques should also be of interest towards addressing the smoothed
complexity of FLIP for max--cut in complete graphs for larger constants .Comment: 36 page
Recommended from our members
Historical sociology of the city
About the book: This Handbook consists of 26 chapters on historical sociology. It is divided into three parts. Part One is devoted to Foundations and covers Marx, Weber, evolutionary and functionalist approaches, the Annales School, Elias, Nelson and Eisenstadt. Part Two moves on to consider major approaches, such as modernization approaches, late Marxist approaches, historical geography, institutional approaches, cultural history, intellectual history, postcolonial and genealogical approaches. The third part is devoted to the major substantive themes in historical sociology ranging from state formation, nationalism, social movements, classes, patriarchy, architecture, religion and moral regulation to problems of periodization and East-West divisions. Each part includes an introduction that summarizes and contextualizes chapters. A general introduction to the volume outlines the current situation of historical sociology after the cultural turn in the social sciences. It argues that historical sociology is deeply divided between explanatory `sociological' approaches and more empirical and interpretative `historical' approaches
Helicopter tail rotor orthogonal blade vortex interaction
The aerodynamic operating environment of the helicopter is particularly complex and,
to some extent, dominated by the vortices trailed from the main and tail rotors. These
vortices not only determine the form of the induced flow field but also interact with
each other and with elements of the physical structure of the flight vehicle. Such
interactions can have implications in terms of structural vibration, noise generation
and flight performance. In this paper, the interaction of main rotor vortices with the
helicopter tail rotor is considered and, in particular, the limiting case of the orthogonal
interaction. The significance of the topic is introduced by highlighting the operational
issues for helicopters arising from tail rotor interactions. The basic phenomenon is
then described before experimental studies of the interaction are presented. Progress
in numerical modelling is then considered and, finally, the prospects for future
research in the area are discussed
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
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