339 research outputs found
Hypergraphs and hypermatrices with symmetric spectrum
It is well known that a graph is bipartite if and only if the spectrum of its
adjacency matrix is symmetric. In the present paper, this assertion is
dissected into three separate matrix results of wider scope, which are extended
also to hypermatrices. To this end the concept of bipartiteness is generalized
by a new monotone property of cubical hypermatrices, called odd-colorable
matrices. It is shown that a nonnegative symmetric -matrix has a
symmetric spectrum if and only if is even and is odd-colorable. This
result also solves a problem of Pearson and Zhang about hypergraphs with
symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.
Separately, similar results are obtained for the -spectram of
hypermatrices.Comment: 17 pages. Corrected proof on p. 1
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
Radio Resource Allocation for Device-to-Device Underlay Communication Using Hypergraph Theory
Device-to-Device (D2D) communication has been recognized as a promising
technique to offload the traffic for the evolved Node B (eNB). However, the D2D
transmission as an underlay causes severe interference to both the cellular and
other D2D links, which imposes a great technical challenge to radio resource
allocation. Conventional graph based resource allocation methods typically
consider the interference between two user equipments (UEs), but they cannot
model the interference from multiple UEs to completely characterize the
interference. In this paper, we study channel allocation using hypergraph
theory to coordinate the interference between D2D pairs and cellular UEs, where
an arbitrary number of D2D pairs are allowed to share the uplink channels with
the cellular UEs. Hypergraph coloring is used to model the cumulative
interference from multiple D2D pairs, and thus, eliminate the mutual
interference. Simulation results show that the system capacity is significantly
improved using the proposed hypergraph method in comparison to the conventional
graph based one.Comment: 27 pages,10 figure
A characterization and an application of weight-regular partitions of graphs
A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex a weight that equals the
corresponding entry of the Perron eigenvector . This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved
Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors
The celebrated Marchenko-Pastur theorem gives the asymptotic spectral
distribution of sums of random, independent, rank-one projections. Its main
hypothesis is that these projections are more or less uniformly distributed on
the first grassmannian, which implies for example that the corresponding
vectors are delocalized, i.e. are essentially supported by the whole canonical
basis. In this paper, we propose a way to drop this delocalization assumption
and we generalize this theorem to a quite general framework, including random
projections whose corresponding vectors are localized, i.e. with some
components much larger than the other ones. The first of our two main examples
is given by heavy tailed random vectors (as in a model introduced by Ben Arous
and Guionnet or as in a model introduced by Zakharevich where the moments grow
very fast as the dimension grows). Our second main example is given by vectors
which are distributed as the Brownian motion on the unit sphere, with localized
initial law. Our framework is in fact general enough to get new correspondences
between classical infinitely divisible laws and some limit spectral
distributions of random matrices, generalizing the so-called Bercovici-Pata
bijection.Comment: 40 pages, 10 figures, some minor mistakes correcte
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