3,673 research outputs found
On the Computational Complexity of Vertex Integrity and Component Order Connectivity
The Weighted Vertex Integrity (wVI) problem takes as input an -vertex
graph , a weight function , and an integer . The
task is to decide if there exists a set such that the weight
of plus the weight of a heaviest component of is at most . Among
other results, we prove that:
(1) wVI is NP-complete on co-comparability graphs, even if each vertex has
weight ;
(2) wVI can be solved in time;
(3) wVI admits a kernel with at most vertices.
Result (1) refutes a conjecture by Ray and Deogun and answers an open
question by Ray et al. It also complements a result by Kratsch et al., stating
that the unweighted version of the problem can be solved in polynomial time on
co-comparability graphs of bounded dimension, provided that an intersection
model of the input graph is given as part of the input.
An instance of the Weighted Component Order Connectivity (wCOC) problem
consists of an -vertex graph , a weight function ,
and two integers and , and the task is to decide if there exists a set
such that the weight of is at most and the weight of
a heaviest component of is at most . In some sense, the wCOC problem
can be seen as a refined version of the wVI problem. We prove, among other
results, that:
(4) wCOC can be solved in time on interval graphs,
while the unweighted version can be solved in time on this graph
class;
(5) wCOC is W[1]-hard on split graphs when parameterized by or by ;
(6) wCOC can be solved in time;
(7) wCOC admits a kernel with at most vertices.
We also show that result (6) is essentially tight by proving that wCOC cannot
be solved in time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the
conference proceedings of ISAAC 201
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
Transport and dynamics on open quantum graphs
We study the classical limit of quantum mechanics on graphs by introducing a
Wigner function for graphs. The classical dynamics is compared to the quantum
dynamics obtained from the propagator. In particular we consider extended open
graphs whose classical dynamics generate a diffusion process. The transport
properties of the classical system are revealed in the scattering resonances
and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
The Transverse-Field Ising Spin Glass Model on the Bethe Lattice with an Application to Adiabatic Quantum Computing
In this Ph.D. thesis we examine the Adiabatic Quantum Algorithm from the point of view of statistical and condensed matter physics. We do this by studying the transverse-field Ising spin glass model defined on the Bethe lattice, which is of independent interest to both the physics community and the quantum computation community.
Using quantum Monte Carlo methods, we perform an extensive study of the the ground-state properties of the model, including the R\'enyi entanglement entropy, quantum Fisher information, Edwards--Anderson parameter, correlation functions.
Through the finite-size scaling of these quantities we find multiple independent and coinciding estimates for the critical point of the glassy phase transition at zero temperature, which completes the phase diagram of the model as was previously known in the literature. We find volumetric bipartite and finite multipartite entanglement for all values of the transverse field considered, both in the paramagnetic and in the glassy phase, and at criticality. We discuss their implication with respect to quantum computing.
By writing a perturbative expansion in the large transverse field regime we develop a mean-field quasiparticle theory that explains the numerical data. The emerging picture is that of degenerate bands of localized quasiparticle excitations on top of a vacuum. The perturbative energy corrections to these bands are given by pair creation/annihilation and hopping processes of the quasiparticles on the Bethe lattice. The transition to the glassy phase is explained as a crossing of the energy level of the vacuum with one of the bands, so that creation of quasiparticles becomes energetically favoured.
We also study the localization properties of the model by employing the forward scattering approximation of the locator expansion, which we compute using a numerical transfer matrix technique. We obtain a lower bound for the mobility edge of the system. We find a localized region inside of the glassy phase and we discuss the consequences of its presence for the Adiabatic Quantum Algorithm
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Quantum fluctuations around low-dimensional topological defects
In these Lectures a method is described to analyze the effect of quantum
fluctuations on topological defect backgrounds up to the one-loop level. The
method is based on the spectral heat kernel/zeta function regularization
procedure, and it is first applied to various types of kinks arising in several
deformed linear and non-linear sigma models with different numbers of scalar
fields. In the second part, the same conceptual framework is constructed for
the topological solitons of the planar semilocal Abelian Higgs model, built
from a doublet of complex scalar fields and one U(1) gauge field.Comment: 63 pages, 14 figures, expanded version of two lectures given by
J.M.G. in 5th International School on Field Theory and Gravitation, Cuiaba,
Brazi
- …