143 research outputs found
Cut-Matching Games on Directed Graphs
We give O(log^2 n)-approximation algorithm based on the cut-matching
framework of [10, 13, 14] for computing the sparsest cut on directed graphs.
Our algorithm uses only O(log^2 n) single commodity max-flow computations and
thus breaks the multicommodity-flow barrier for computing the sparsest cut on
directed graph
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
Many Sparse Cuts via Higher Eigenvalues
Cheeger's fundamental inequality states that any edge-weighted graph has a
vertex subset such that its expansion (a.k.a. conductance) is bounded as
follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}}
\leq 2\sqrt{\lambda_2} where is the total edge weight of a subset or a
cut and is the second smallest eigenvalue of the normalized
Laplacian of the graph. Here we prove the following natural generalization: for
any integer , there exist disjoint subsets ,
such that where
is the smallest eigenvalue of the normalized Laplacian and
are suitable absolute constants. Our proof is via a polynomial-time
algorithm to find such subsets, consisting of a spectral projection and a
randomized rounding. As a consequence, we get the same upper bound for the
small set expansion problem, namely for any , there is a subset whose
weight is at most a \bigO(1/k) fraction of the total weight and . Both results are the best possible up to constant
factors.
The underlying algorithmic problem, namely finding subsets such that the
maximum expansion is minimized, besides extending sparse cuts to more than one
subset, appears to be a natural clustering problem in its own right
Partitioning into Expanders
Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue
of the normalized laplacian matrix of G. There is a basic fact in algebraic
graph theory that lambda_k > 0 if and only if G has at most k-1 connected
components. We prove a robust version of this fact. If lambda_k>0, then for
some 1\leq \ell\leq k-1, V can be {\em partitioned} into l sets P_1,\ldots,P_l
such that each P_i is a low-conductance set in G and induces a high conductance
induced subgraph. In particular, \phi(P_i)=O(l^3\sqrt{\lambda_l}) and
\phi(G[P_i]) >= \lambda_k/k^2).
We make our results algorithmic by designing a simple polynomial time
spectral algorithm to find such partitioning of G with a quadratic loss in the
inside conductance of P_i's. Unlike the recent results on higher order
Cheeger's inequality [LOT12,LRTV12], our algorithmic results do not use higher
order eigenfunctions of G. If there is a sufficiently large gap between
lambda_k and lambda_{k+1}, more precisely, if \lambda_{k+1} >= \poly(k)
lambda_{k}^{1/4} then our algorithm finds a k partitioning of V into sets
P_1,...,P_k such that the induced subgraph G[P_i] has a significantly larger
conductance than the conductance of P_i in G. Such a partitioning may represent
the best k clustering of G. Our algorithm is a simple local search that only
uses the Spectral Partitioning algorithm as a subroutine. We expect to see
further applications of this simple algorithm in clustering applications
Approximating Non-Uniform Sparsest Cut via Generalized Spectra
We give an approximation algorithm for non-uniform sparsest cut with the
following guarantee: For any , given cost and demand
graphs with edge weights respectively, we can find a set
with at most
times the optimal non-uniform sparsest cut value,
in time 2^{r/(\delta\epsilon)}\poly(n) provided . Here is the 'th smallest generalized
eigenvalue of the Laplacian matrices of cost and demand graphs; (resp. ) is the weight of edges crossing the
cut in cost (resp. demand) graph and is the
sparsity of the optimal cut. In words, we show that the non-uniform sparsest
cut problem is easy when the generalized spectrum grows moderately fast. To the
best of our knowledge, there were no results based on higher order spectra for
non-uniform sparsest cut prior to this work.
Even for uniform sparsest cut, the quantitative aspects of our result are
somewhat stronger than previous methods. Similar results hold for other
expansion measures like edge expansion, normalized cut, and conductance, with
the 'th smallest eigenvalue of the normalized Laplacian playing the role of
in the latter two cases.
Our proof is based on an l1-embedding of vectors from a semi-definite program
from the Lasserre hierarchy. The embedded vectors are then rounded to a cut
using standard threshold rounding. We hope that the ideas connecting
-embeddings to Lasserre SDPs will find other applications. Another
aspect of the analysis is the adaptation of the column selection paradigm from
our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather
than vertices. This feature is important in order to extend the algorithms to
non-uniform sparsest cut.Comment: 16 page
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