63,874 research outputs found
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width
Fourier sparsity, spectral norm, and the Log-rank conjecture
We study Boolean functions with sparse Fourier coefficients or small spectral
norm, and show their applications to the Log-rank Conjecture for XOR functions
f(x\oplus y) --- a fairly large class of functions including well studied ones
such as Equality and Hamming Distance. The rank of the communication matrix M_f
for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree
of f and D^CC(f) stand for the deterministic communication complexity for
f(x\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\hat f||_1). In
particular, the Log-rank conjecture holds for XOR functions with constant
F2-degree. 2. D^CC(f) = O(d ||\hat f||_1) = O(\sqrt{rank(M_f)}\logrank(M_f)).
We obtain our results through a degree-reduction protocol based on a variant of
polynomial rank, and actually conjecture that its communication cost is already
\log^{O(1)}rank(M_f). The above bounds also hold for the parity decision tree
complexity of f, a measure that is no less than the communication complexity
(up to a factor of 2).
Along the way we also show several structural results about Boolean functions
with small F2-degree or small spectral norm, which could be of independent
interest. For functions f with constant F2-degree: 1) f can be written as the
summation of quasi-polynomially many indicator functions of subspaces with
\pm-signs, improving the previous doubly exponential upper bound by Green and
Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having
a small parity decision tree complexity; 3) f depends only on polylog||\hat
f||_1 linear functions of input variables. For functions f with small spectral
norm: 1) there is an affine subspace with co-dimension O(||\hat f||_1) on which
f is a constant; 2) there is a parity decision tree with depth O(||\hat f||_1
log ||\hat f||_0).Comment: v2: Corollary 31 of v1 removed because of a bug in the proof. (Other
results not affected.
Vertex decompositions of two-dimensional complexes and graphs
We investigate families of two-dimensional simplicial complexes defined in
terms of vertex decompositions. They include nonevasive complexes, strongly
collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary
and Palmer. We investigate the complexity of recognition problems for those
families and some of their combinatorial properties. Certain results follow
from analogous decomposition techniques for graphs. For example, we prove that
it is NP-complete to decide if a graph can be reduced to a discrete graph by a
sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug
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