645 research outputs found
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
Maintaining Expander Decompositions via Sparse Cuts
In this article, we show that the algorithm of maintaining expander
decompositions in graphs undergoing edge deletions directly by removing sparse
cuts repeatedly can be made efficient. Formally, for an -edge undirected
graph , we say a cut is -sparse if . A
-expander decomposition of is a partition of into sets such that each cluster contains no -sparse cut
(meaning it is a -expander) with edges crossing
between clusters. A natural way to compute a -expander decomposition is
to decompose clusters by -sparse cuts until no such cut is contained in
any cluster. We show that even in graphs undergoing edge deletions, a slight
relaxation of this meta-algorithm can be implemented efficiently with amortized
update time . Our approach naturally extends to maintaining
directed -expander decompositions and -expander hierarchies and
thus gives a unifying framework while having simpler proofs than previous
state-of-the-art work. In all settings, our algorithm matches the run-times of
previous algorithms up to subpolynomial factors. Moreover, our algorithm
provides stronger guarantees for -expander decompositions. For example,
for graphs undergoing edge deletions, our approach is the first to maintain a
dynamic expander decomposition where each updated decomposition is a refinement
of the previous decomposition, and our approach is the first to guarantee a
sublinear bound on the total number of edges that cross
between clusters across the entire sequence of dynamic updates
Expander Decomposition in Dynamic Streams
In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?
Expander Decomposition in Dynamic Streams
In this paper we initiate the study of expander decompositions of a graph
in the streaming model of computation. The goal is to find a
partitioning of vertices such that the subgraphs of
induced by the clusters are good expanders, while the
number of intercluster edges is small. Expander decompositions are classically
constructed by a recursively applying balanced sparse cuts to the input graph.
In this paper we give the first implementation of such a recursive sparsest cut
process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as
a power cut sparsifier - it preserves cuts in any given vertex induced subgraph
(or, any cluster in a fixed partition of ) to within a -multiplicative/additive error with high probability. The power cut
sparsifier uses space and edges, which we show is
asymptotically tight up to polylogarithmic factors in for constant
.Comment: 31 pages, 0 figures, to appear in ITCS 202
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