28 research outputs found
Approximating the Expansion Profile and Almost Optimal Local Graph Clustering
Spectral partitioning is a simple, nearly-linear time, algorithm to find
sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for
the quality of the approximation found by the algorithm. Local graph
partitioning algorithms [ST08,ACL06,AP09] run in time that is nearly linear in
the size of the output set, and their approximation guarantee is worse than the
guarantee provided by the Cheeger inequalities by a polylogarithmic
factor. It has been a long standing open problem to design
a local graph clustering algorithm with an approximation guarantee close to the
guarantee of the Cheeger inequalities and with a running time nearly linear in
the size of the output.
In this paper we solve this problem; we design an algorithm with the same
guarantee (up to a constant factor) as the Cheeger inequality, that runs in
time slightly super linear in the size of the output. This is the first
sublinear (in the size of the input) time algorithm with almost the same
guarantee as the Cheeger's inequality. As a byproduct of our results, we prove
a bicriteria approximation algorithm for the expansion profile of any graph.
Let . There is a polynomial
time algorithm that, for any , finds a set of measure
, and expansion . Our proof techniques also provide a simpler
proof of the structural result of Arora, Barak, Steurer [ABS10], that can be
applied to irregular graphs.
Our main technical tool is that for any set of vertices of a graph, a
lazy -step random walk started from a randomly chosen vertex of , will
remain entirely inside with probability at least . This
itself provides a new lower bound to the uniform mixing time of any finite
states reversible markov chain
Gap Amplification for Small-Set Expansion via Random Walks
In this work, we achieve gap amplification for the Small-Set Expansion
problem. Specifically, we show that an instance of the Small-Set Expansion
Problem with completeness and soundness is at least as
difficult as Small-Set Expansion with completeness and soundness
, for any function which grows faster than
. We achieve this amplification via random walks -- our gadget
is the graph with adjacency matrix corresponding to a random walk on the
original graph. An interesting feature of our reduction is that unlike gap
amplification via parallel repetition, the size of the instances (number of
vertices) produced by the reduction remains the same
Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure
We study the problem of finding and characterizing subgraphs with small
\textit{bipartiteness ratio}. We give a bicriteria approximation algorithm
\verb|SwpDB| such that if there exists a subset of volume at most and
bipartiteness ratio , then for any , it finds a set
of volume at most and bipartiteness ratio at most
. By combining a truncation operation, we give a local
algorithm \verb|LocDB|, which has asymptotically the same approximation
guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness
ratio of the output set, and runs in time
, independent of the size of the
graph. Finally, we give a spectral characterization of the small dense
bipartite-like subgraphs by using the th \textit{largest} eigenvalue of the
Laplacian of the graph.Comment: 17 pages; ISAAC 201
Testing Small Set Expansion in General Graphs
We consider the problem of testing small set expansion for general graphs. A
graph is a -expander if every subset of volume at most has
conductance at least . Small set expansion has recently received
significant attention due to its close connection to the unique games
conjecture, the local graph partitioning algorithms and locally testable codes.
We give testers with two-sided error and one-sided error in the adjacency
list model that allows degree and neighbor queries to the oracle of the input
graph. The testers take as input an -vertex graph , a volume bound ,
an expansion bound and a distance parameter . For the
two-sided error tester, with probability at least , it accepts the graph
if it is a -expander and rejects the graph if it is -far
from any -expander, where and
. The
query complexity and running time of the tester are
, where is the number of
edges of the graph. For the one-sided error tester, it accepts every
-expander, and with probability at least , rejects every graph
that is -far from -expander, where
and for any . The query
complexity and running time of this tester are
.
We also give a two-sided error tester with smaller gap between and
in the rotation map model that allows (neighbor, index) queries and
degree queries.Comment: 23 pages; STACS 201