1,495 research outputs found
Permutation-partition pairs. III. Embedding distributions of linear families of graphs
AbstractFor any fixed graph H, and H-linear family of graphs is a sequence {Gn}n=1∞ of graphs in which Gn consists of n copies of H that have been linked in a consistent manner so as to form a chain. Generating functions for the region distribution of any such family are found. It is also shown that the minimum genus and the average genus of Gn are essentially linear functions of n
Random 2-cell embeddings of multistars
Random 2-cell embeddings of a given graph are obtained by choosing a
random local rotation around every vertex. We analyze the expected number of
faces, , of such an embedding which is equivalent to studying
its average genus. So far, tight results are known for two families called
monopoles and dipoles. We extend the dipole result to a more general family
called multistars, i.e., loopless multigraphs in which there is a vertex
incident with all the edges. In particular, we show that the expected number of
faces of every multistar with nonleaf edges lies in an interval of length
centered at the expected number of faces of an -edge dipole.
This allows us to derive bounds on for any given graph in
terms of vertex degrees. We conjecture that for any
simple -vertex graph .Comment: 15 pages, 2 figures. Accepted to European conference on
combinatorics, graph theory and applications (EUROCOMB) 202
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Harmonic functions on multiplicative graphs and interpolation polynomials
We construct examples of nonnegative harmonic functions on certain graded
graphs: the Young lattice and its generalizations. Such functions first emerged
in harmonic analysis on the infinite symmetric group. Our method relies on
multivariate interpolation polynomials associated with Schur's S and P
functions and with Jack symmetric functions. As a by-product, we compute
certain Selberg-type integrals.Comment: AMSTeX, 35 page
Compound real Wishart and q-Wishart matrices
We introduce a family of matrices with non-commutative entries that
generalize the classical real Wishart matrices.
With the help of the Brauer product, we derive a non-asymptotic expression
for the moments of traces of monomials in such matrices; the expression is
quite similar to the formula derived in our previous work for independent
complex Wishart matrices. We then analyze the fluctuations about the
Marchenko-Pastur law. We show that after centering by the mean, traces of real
symmetric polynomials in q-Wishart matrices converge in distribution, and we
identify the asymptotic law as the normal law when q=1, and as the semicircle
law when q=0
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
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