1,025 research outputs found
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
We consider sequences of graphs and define various notions of convergence
related to these sequences: ``left convergence'' defined in terms of the
densities of homomorphisms from small graphs into the graphs of the sequence,
and ``right convergence'' defined in terms of the densities of homomorphisms
from the graphs of the sequence into small graphs; and convergence in a
suitably defined metric.
In Part I of this series, we show that left convergence is equivalent to
convergence in metric, both for simple graphs, and for graphs with nodeweights
and edgeweights. One of the main steps here is the introduction of a
cut-distance comparing graphs, not necessarily of the same size. We also show
how these notions of convergence provide natural formulations of Szemeredi
partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This
version differs from an earlier version from May 2006 in the organization of
the sections, but is otherwise almost identica
Finding the Minimum-Weight k-Path
Given a weighted -vertex graph with integer edge-weights taken from a
range , we show that the minimum-weight simple path visiting
vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k
M). If the weights are reals in , we provide a
-approximation which has a running time of \tilde{O}(2^k
\poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem
of -tree, in which we wish to find a minimum-weight copy of a -node tree
in a given weighted graph , under the same restrictions on edge weights
respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k)
M n^3) and a -approximate solution of running time
\tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above
algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201
Resource theories of knowledge
How far can we take the resource theoretic approach to explore physics?
Resource theories like LOCC, reference frames and quantum thermodynamics have
proven a powerful tool to study how agents who are subject to certain
constraints can act on physical systems. This approach has advanced our
understanding of fundamental physical principles, such as the second law of
thermodynamics, and provided operational measures to quantify resources such as
entanglement or information content. In this work, we significantly extend the
approach and range of applicability of resource theories. Firstly we generalize
the notion of resource theories to include any description or knowledge that
agents may have of a physical state, beyond the density operator formalism. We
show how to relate theories that differ in the language used to describe
resources, like micro and macroscopic thermodynamics. Finally, we take a
top-down approach to locality, in which a subsystem structure is derived from a
global theory rather than assumed. The extended framework introduced here
enables us to formalize new tasks in the language of resource theories, ranging
from tomography, cryptography, thermodynamics and foundational questions, both
within and beyond quantum theory.Comment: 28 pages featuring figures, examples, map and neatly boxed theorems,
plus appendi
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