8,346 research outputs found
Computational Complexity of the Interleaving Distance
The interleaving distance is arguably the most prominent distance measure in
topological data analysis. In this paper, we provide bounds on the
computational complexity of determining the interleaving distance in several
settings. We show that the interleaving distance is NP-hard to compute for
persistence modules valued in the category of vector spaces. In the specific
setting of multidimensional persistent homology we show that the problem is at
least as hard as a matrix invertibility problem. Furthermore, this allows us to
conclude that the interleaving distance of interval decomposable modules
depends on the characteristic of the field. Persistence modules valued in the
category of sets are also studied. As a corollary, we obtain that the
isomorphism problem for Reeb graphs is graph isomorphism complete.Comment: Discussion related to the characteristic of the field added. Paper
accepted to the 34th International Symposium on Computational Geometr
Computing the interleaving distance is NP-hard
We show that computing the interleaving distance between two multi-graded
persistence modules is NP-hard. More precisely, we show that deciding whether
two modules are -interleaved is NP-complete, already for bigraded, interval
decomposable modules. Our proof is based on previous work showing that a
constrained matrix invertibility problem can be reduced to the interleaving
distance computation of a special type of persistence modules. We show that
this matrix invertibility problem is NP-complete. We also give a slight
improvement of the above reduction, showing that also the approximation of the
interleaving distance is NP-hard for any approximation factor smaller than .
Additionally, we obtain corresponding hardness results for the case that the
modules are indecomposable, and in the setting of one-sided stability.
Furthermore, we show that checking for injections (resp. surjections) between
persistence modules is NP-hard. In conjunction with earlier results from
computational algebra this gives a complete characterization of the
computational complexity of one-sided stability. Lastly, we show that it is in
general NP-hard to approximate distances induced by noise systems within a
factor of 2.Comment: 25 pages. Several expository improvements and minor corrections. Also
added a section on noise system
Kaluza-Klein description of geometric phases in graphene
In this paper, we use the Kaluza-Klein approach to describe topological
defects in a graphene layer. Using this approach, we propose a geometric model
allowing to discuss the quantum flux in -spin subspace. Within this model,
the graphene layer with a topological defect is described by a four-dimensional
metric, where the deformation produced by the topological defect is introduced
via the three-dimensional part of metric tensor, while an Abelian gauge field
is introduced via an extra dimension. We use this new geometric model to
discuss the arising of topological quantum phases in a graphene layer with a
topological defect.Comment: 16 pages, version accepted to Annals of Physic
Abelian geometric phase for a Dirac neutral particle in a Lorentz symmetry violation environment
We introduce a new term into the Dirac equation based on the Lorentz symmetry
violation background in order to make a theoretical description of the
relativistic quantum dynamics of a spin-half neutral particle, where the wave
function of the neutral particle acquires a relativistic Abelian quantum phase
given by the interaction between a fixed time-like 4-vector background and
crossed electric and magnetic fields, which is analogous to the geometric phase
obtained by Wei \textit{et al} [H. Wei, R. Han and X. Wei, Phys. Rev. Lett.
\textbf{75}, 2071 (1995)] for a spinless neutral particle with an induced
electric dipole moment. We also discuss the flux dependence of energy levels of
bound states analogous to the Aharonov-Bohm effect for bound states.Comment: 16 pages, no figure
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