506 research outputs found
Network Capacity Bound for Personalized PageRank in Multimodal Networks
In a former paper the concept of Bipartite PageRank was introduced and a
theorem on the limit of authority flowing between nodes for personalized
PageRank has been generalized. In this paper we want to extend those results to
multimodal networks. In particular we introduce a hypergraph type that may be
used for describing multimodal network where a hyperlink connects nodes from
each of the modalities. We introduce a generalisation of PageRank for such
graphs and define the respective random walk model that can be used for
computations. we finally state and prove theorems on the limit of outflow of
authority for cases where individual modalities have identical and distinct
damping factors.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1702.0373
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
A simple and sharper proof of the hypergraph Moore bound
The hypergraph Moore bound is an elegant statement that characterizes the
extremal trade-off between the girth - the number of hyperedges in the smallest
cycle or even cover (a subhypergraph with all degrees even) and size - the
number of hyperedges in a hypergraph. For graphs (i.e., -uniform
hypergraphs), a bound tight up to the leading constant was proven in a
classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity
, an appropriate generalization was conjectured by Feige [Fei08]. The
conjecture was settled up to an additional factor in the size
in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument
relies on a connection between the existence of short even covers and the
spectrum of a certain randomly signed Kikuchi matrix. Their analysis,
especially for the case of odd , is significantly complicated.
In this work, we present a substantially simpler and shorter proof of the
hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi
matrix and an edge deletion step that allows us to drop several involved steps
in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi
matrix and the use of the Schudy-Sviridenko polynomial concentration. Our
simpler proof also obtains tighter parameters: in particular, the argument
gives a new proof of the classical Moore bound of [AHL02] with no loss (the
proof in [GKM21] loses a factor), and loses only a single
logarithmic factor for all -uniform hypergraphs.
As in [GKM21], our ideas naturally extend to yield a simpler proof of the
full trade-off for strongly refuting smoothed instances of constraint
satisfaction problems with similarly improved parameters
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
On Hamiltonian cycles in hypergraphs with dense link graphs
We show that every -uniform hypergraph on vertices whose minimum
-degree is at least contains a Hamiltonian cycle. A
construction due to Han and Zhao shows that this minimum degree condition is
optimal. The same result was proved independently by Lang and Sahueza-Matamala.Comment: Dedicated to Endre Szemer\'edi on the occasion of his 80th birthda
Continuous-time quantum walks on dynamical percolation graphs
We address continuous-time quantum walks on graphs in the presence of time-
and space-dependent noise. Noise is modeled as generalized dynamical
percolation, i.e. classical time-dependent fluctuations affecting the tunneling
amplitudes of the walker. In order to illustrate the general features of the
model, we review recent results on two paradigmatic examples: the dynamics of
quantum walks on the line and the effects of noise on the performances of
quantum spatial search on the complete and the star graph. We also discuss
future perspectives, including extension to many-particle quantum walk, to
noise model for on-site energies and to the analysis of different noise
spectra. Finally, we address the use of quantum walks as a quantum probe to
characterize defects and perturbations occurring in complex, classical and
quantum, networks.Comment: 7 pages, 4 figures. Accepted for publication in EPL Perspective
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