## Finding Sparse Cuts Locally Using Evolving Sets

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Andersen, Reid

Peres, Yuval

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A {\em local graph partitioning algorithm} finds a set of vertices with small
conductance (i.e. a sparse cut) by adaptively exploring part of a large graph
$G$, starting from a specified vertex. For the algorithm to be local, its
complexity must be bounded in terms of the size of the set that it outputs,
with at most a weak dependence on the number $n$ of vertices in $G$. Previous
local partitioning algorithms find sparse cuts using random walks and
personalized PageRank. In this paper, we introduce a randomized local
partitioning algorithm that finds a sparse cut by simulating the {\em
volume-biased evolving set process}, which is a Markov chain on sets of
vertices. We prove that for any set of vertices $A$ that has conductance at
most $\phi$, for at least half of the starting vertices in $A$ our algorithm
will output (with probability at least half), a set of conductance
$O(\phi^{1/2} \log^{1/2} n)$. We prove that for a given run of the algorithm,
the expected ratio between its computational complexity and the volume of the
set that it outputs is $O(\phi^{-1/2} polylog(n))$. In comparison, the best
previous local partitioning algorithm, due to Andersen, Chung, and Lang, has
the same approximation guarantee, but a larger ratio of $O(\phi^{-1}
polylog(n))$ between the complexity and output volume. Using our local
partitioning algorithm as a subroutine, we construct a fast algorithm for
finding balanced cuts. Given a fixed value of $\phi$, the resulting algorithm
has complexity $O((m+n\phi^{-1/2}) polylog(n))$ and returns a cut with
conductance $O(\phi^{1/2} \log^{1/2} n)$ and volume at least $v_{\phi}/2$,
where $v_{\phi}$ is the largest volume of any set with conductance at most
$\phi$.

Comment: 20 pages, no figures

Comment: 20 pages, no figures

##### Keywords

Computer Science - Data Structures and Algorithms, F.2.2