590 research outputs found
Constant Factor Approximation for Balanced Cut in the PIE model
We propose and study a new semi-random semi-adversarial model for Balanced
Cut, a planted model with permutation-invariant random edges (PIE). Our model
is much more general than planted models considered previously. Consider a set
of vertices V partitioned into two clusters and of equal size. Let
be an arbitrary graph on with no edges between and . Let
be a set of edges sampled from an arbitrary permutation-invariant
distribution (a distribution that is invariant under permutation of vertices in
and in ). Then we say that is a graph with
permutation-invariant random edges.
We present an approximation algorithm for the Balanced Cut problem that finds
a balanced cut of cost in this model.
In the regime when , this is a
constant factor approximation with respect to the cost of the planted cut.Comment: Full version of the paper at the 46th ACM Symposium on the Theory of
Computing (STOC 2014). 32 page
Monotone Maps, Sphericity and Bounded Second Eigenvalue
We consider {\em monotone} embeddings of a finite metric space into low
dimensional normed space. That is, embeddings that respect the order among the
distances in the original space. Our main interest is in embeddings into
Euclidean spaces. We observe that any metric on points can be embedded into
, while, (in a sense to be made precise later), for almost every
-point metric space, every monotone map must be into a space of dimension
.
It becomes natural, then, to seek explicit constructions of metric spaces
that cannot be monotonically embedded into spaces of sublinear dimension. To
this end, we employ known results on {\em sphericity} of graphs, which suggest
one example of such a metric space - that defined by a complete bipartitegraph.
We prove that an -regular graph of order , with bounded diameter
has sphericity , where is the second
largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq
\half is constant. We also show that while random graphs have linear
sphericity, there are {\em quasi-random} graphs of logarithmic sphericity.
For the above bound to be linear, must be constant. We show that
if the second eigenvalue of an -regular graph is bounded by a constant,
then the graph is close to being complete bipartite. Namely, its adjacency
matrix differs from that of a complete bipartite graph in only
entries. Furthermore, for any 0 < \delta < \half, and , there are
only finitely many -regular graphs with second eigenvalue at most
Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values
Let X be an algebraic curve over Q and t a non-constant Q-rational function
on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a
point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the
number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this
conjecture in the special case when t defines a geometrically abelian covering
of the projective line, and the critical values of t are all rational. This
implies, in particular, that our conjecture follows from a famous conjecture of
Schinzel.Comment: Some typos are corrected. The article is now accepted in Periodica
Math. Hungaric
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