46,058 research outputs found
Boolean Matrix Factorization Meets Consecutive Ones Property
Boolean matrix factorization is a natural and a popular technique for summarizing binary matrices. In this paper, we study a problem of Boolean matrix factorization where we additionally require that the factor matrices have consecutive ones property (OBMF). A major application of this optimization problem comes from graph visualization: standard techniques for visualizing graphs are circular or linear layout, where nodes are ordered in circle or on a line. A common problem with visualizing graphs is clutter due to too many edges. The standard approach to deal with this is to bundle edges together and represent them as ribbon. We also show that we can use OBMF for edge bundling combined with circular or linear layout techniques. We demonstrate that not only this problem is NP-hard but we cannot have a polynomial-time algorithm that yields a multiplicative approximation guarantee (unless P = NP). On the positive side, we develop a greedy algorithm where at each step we look for the best 1-rank factorization. Since even obtaining 1-rank factorization is NP-hard, we propose an iterative algorithm where we fix one side and and find the other, reverse the roles, and repeat. We show that this step can be done in linear time using pq-trees. We also extend the problem to cyclic ones property and symmetric factorizations. Our experiments show that our algorithms find high-quality factorizations and scale well
Spin-1 effective Hamiltonian with three degenerate orbitals: An application to the case of V_2O_3
Motivated by recent neutron and x-ray observations in V_2O_3, we derive the
effective Hamiltonian in the strong coupling limit of an Hubbard model with
three degenerate t_{2g} states containing two electrons coupled to spin S = 1,
and use it to re-examine the low-temperature ground-state properties of this
compound. An axial trigonal distortion of the cubic states is also taken into
account. Since there are no assumptions about the symmetry properties of the
hopping integrals involved, the resulting spin-orbital Hamiltonian can be
generally applied to any crystallographic configuration of the transition metal
ion giving rise to degenerate t_{2g} orbitals. Specializing to the case of
V_2O_3 we consider the antiferromagnetic insulating phase. We find two
variational regimes, depending on the relative size of the correlation energy
of the vertical pairs and the in-plane interaction energy. The former favors
the formation of stable molecules throughout the crystal, while the latter
tends to break this correlated state. We determine in both cases the minimizing
orbital solutions for various spin configurations, and draw the corresponding
phase diagrams. We find that none of the symmetry-breaking stable phases with
the real spin structure presents an orbital ordering compatible with the
magnetic space group indicated by very recent observations of non-reciprocal
x-ray gyrotropy in V_2O_3. We do however find a compatible solution with very
small excitation energy in two distinct regions of the phase space, which might
turn into the true ground state of V_2O_3 due to the favorable coupling with
the lattice. We illustrate merits and drawbacks of the various solutions and
discuss them in relation to the present experimental evidence.Comment: 36 pages, 19 figure
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
Variations on the Seventh Route to Relativity
As motivated in the full abstract, this paper further investigates Barbour,
Foster and O Murchadha (BFO)'s 3-space formulation of GR. This is based on
best-matched lapse-eliminated actions and gives rise to several theories
including GR and a conformal gravity theory. We study the simplicity postulates
assumed in BFO's work and how to weaken them, so as to permit the inclusion of
the full set of matter fields known to occur in nature.
We study the configuration spaces of gravity-matter systems upon which BFO's
formulation leans. In further developments the lapse-eliminated actions used by
BFO become impractical and require generalization. We circumvent many of these
problems by the equivalent use of lapse-uneliminated actions, which furthermore
permit us to interpret BFO's formulation within Kuchar's generally covariant
hypersurface framework. This viewpoint provides alternative reasons to BFO's as
to why the inclusion of bosonic fields in the 3-space approach gives rise to
minimally-coupled scalar fields, electromagnetism and Yang--Mills theory. This
viewpoint also permits us to quickly exhibit further GR-matter theories
admitted by the 3-space formulation. In particular, we show that the spin-1/2
fermions of the theories of Dirac, Maxwell--Dirac and Yang--Mills--Dirac, all
coupled to GR, are admitted by the generalized 3-space formulation we present.
Thus all the known fundamental matter fields can be accommodated. This
corresponds to being able to pick actions for all these theories which have
less kinematics than suggested by the generally covariant hypersurface
framework. For all these theories, Wheeler's thin sandwich conjecture may be
posed, rendering them timeless in Barbour's sense.Comment: Revtex version; Journal-ref adde
Spectral Duality Between Heisenberg Chain and Gaudin Model
In our recent paper we described relationships between integrable systems
inspired by the AGT conjecture. On the gauge theory side an integrable spin
chain naturally emerges while on the conformal field theory side one obtains
some special reduced Gaudin model. Two types of integrable systems were shown
to be related by the spectral duality. In this paper we extend the spectral
duality to the case of higher spin chains. It is proved that the N-site GL(k)
Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model.
Moreover, we construct an explicit Poisson map between the models at the
classical level by performing the Dirac reduction procedure and applying the
AHH duality transformation.Comment: 36 page
Shared Memory Parallel Subgraph Enumeration
The subgraph enumeration problem asks us to find all subgraphs of a target
graph that are isomorphic to a given pattern graph. Determining whether even
one such isomorphic subgraph exists is NP-complete---and therefore finding all
such subgraphs (if they exist) is a time-consuming task. Subgraph enumeration
has applications in many fields, including biochemistry and social networks,
and interestingly the fastest algorithms for solving the problem for
biochemical inputs are sequential. Since they depend on depth-first tree
traversal, an efficient parallelization is far from trivial. Nevertheless,
since important applications produce data sets with increasing difficulty,
parallelism seems beneficial.
We thus present here a shared-memory parallelization of the state-of-the-art
subgraph enumeration algorithms RI and RI-DS (a variant of RI for dense graphs)
by Bonnici et al. [BMC Bioinformatics, 2013]. Our strategy uses work stealing
and our implementation demonstrates a significant speedup on real-world
biochemical data---despite a highly irregular data access pattern. We also
improve RI-DS by pruning the search space better; this further improves the
empirical running times compared to the already highly tuned RI-DS.Comment: 18 pages, 12 figures, To appear at the 7th IEEE Workshop on Parallel
/ Distributed Computing and Optimization (PDCO 2017
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