6 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
Consistency Thresholds for the Planted Bisection Model
The planted bisection model is a random graph model in which the nodes are
divided into two equal-sized communities and then edges are added randomly in a
way that depends on the community membership. We establish necessary and
sufficient conditions for the asymptotic recoverability of the planted
bisection in this model. When the bisection is asymptotically recoverable, we
give an efficient algorithm that successfully recovers it. We also show that
the planted bisection is recoverable asymptotically if and only if with high
probability every node belongs to the same community as the majority of its
neighbors.
Our algorithm for finding the planted bisection runs in time almost linear in
the number of edges. It has three stages: spectral clustering to compute an
initial guess, a "replica" stage to get almost every vertex correct, and then
some simple local moves to finish the job. An independent work by Abbe,
Bandeira, and Hall establishes similar (slightly weaker) results but only in
the case of logarithmic average degree.Comment: latest version contains an erratum, addressing an error pointed out
by Jan van Waai
Constant factor approximation for balanced cut in the PIE model Tasos Sidiropoulos: Spectral concentration, robust k-center, and simple clustering
Non UBCUnreviewedAuthor affiliation: Microsoft ResearchOthe