We give an approximation algorithm for non-uniform sparsest cut with the
following guarantee: For any ϵ,δ∈(0,1), given cost and demand
graphs with edge weights C,D respectively, we can find a set T⊆V
with D(T,V∖T)C(T,V∖T)​ at most
δ1+ϵ​ times the optimal non-uniform sparsest cut value,
in time 2^{r/(\delta\epsilon)}\poly(n) provided λr​≥Φ∗/(1−δ). Here λr​ is the r'th smallest generalized
eigenvalue of the Laplacian matrices of cost and demand graphs; C(T,V∖T) (resp. D(T,V∖T)) is the weight of edges crossing the
(T,V∖T) cut in cost (resp. demand) graph and Φ∗ 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 r'th smallest eigenvalue of the normalized Laplacian playing the role of
λr​ 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
ℓ1​-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