32,425 research outputs found
A Note on a Consistency Property for Permutations
AbstractIn cooperative game theory allocation of earnings to players may take place on the basis of selectors or—more restrictively—consistent selectors, or on the basis of a permutation representing the queueing of the players. This note gives a graph theoretic characterization of those situations in which the latter allocation method results in allocation with consistent selectors
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks
Given a pair of graphs with the same number of vertices, the inexact graph
matching problem consists in finding a correspondence between the vertices of
these graphs that minimizes the total number of induced edge disagreements. We
study this problem from a statistical framework in which one of the graphs is
an errorfully observed copy of the other. We introduce a corrupting channel
model, and show that in this model framework, the solution to the graph
matching problem is a maximum likelihood estimator. Necessary and sufficient
conditions for consistency of this MLE are presented, as well as a relaxed
notion of consistency in which a negligible fraction of the vertices need not
be matched correctly. The results are used to study matchability in several
families of random graphs, including edge independent models, random regular
graphs and small-world networks. We also use these results to introduce
measures of matching feasibility, and experimentally validate the results on
simulated and real-world networks
Protected gates for topological quantum field theories
We study restrictions on locality-preserving unitary logical gates for
topological quantum codes in two spatial dimensions. A locality-preserving
operation is one which maps local operators to local operators --- for example,
a constant-depth quantum circuit of geometrically local gates, or evolution for
a constant time governed by a geometrically-local bounded-strength Hamiltonian.
Locality-preserving logical gates of topological codes are intrinsically fault
tolerant because spatially localized errors remain localized, and hence
sufficiently dilute errors remain correctable. By invoking general properties
of two-dimensional topological field theories, we find that the
locality-preserving logical gates are severely limited for codes which admit
non-abelian anyons; in particular, there are no locality-preserving logical
gates on the torus or the sphere with M punctures if the braiding of anyons is
computationally universal. Furthermore, for Ising anyons on the M-punctured
sphere, locality-preserving gates must be elements of the logical Pauli group.
We derive these results by relating logical gates of a topological code to
automorphisms of the Verlinde algebra of the corresponding anyon model, and by
requiring the logical gates to be compatible with basis changes in the logical
Hilbert space arising from local F-moves and the mapping class group.Comment: 50 pages, many figures, v3: updated to match published versio
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