56 research outputs found
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut
We give improved pseudorandom generators (PRGs) for Lipschitz functions of
low-degree polynomials over the hypercube. These are functions of the form
psi(P(x)), where P is a low-degree polynomial and psi is a function with small
Lipschitz constant. PRGs for smooth functions of low-degree polynomials have
received a lot of attention recently and play an important role in constructing
PRGs for the natural class of polynomial threshold functions. In spite of the
recent progress, no nontrivial PRGs were known for fooling Lipschitz functions
of degree O(log n) polynomials even for constant error rate. In this work, we
give the first such generator obtaining a seed-length of (log
n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps.
Previous generators had an exponential dependence on the degree.
We use our PRG to get better integrality gap instances for sparsest cut, a
fundamental problem in graph theory with many applications in graph
optimization. We give an instance of uniform sparsest cut for which a powerful
semi-definite relaxation (SDP) first introduced by Goemans and Linial and
studied in the seminal work of Arora, Rao and Vazirani has an integrality gap
of exp(\Omega((log log n)^{1/2})). Understanding the performance of the
Goemans-Linial SDP for uniform sparsest cut is an important open problem in
approximation algorithms and metric embeddings and our work gives a
near-exponential improvement over previous lower bounds which achieved a gap of
\Omega(log log n)
Integrality gaps of semidefinite programs for Vertex Cover and relations to embeddability of Negative Type metrics
We study various SDP formulations for {\sc Vertex Cover} by adding different
constraints to the standard formulation. We show that {\sc Vertex Cover} cannot
be approximated better than even when we add the so called pentagonal
inequality constraints to the standard SDP formulation, en route answering an
open question of Karakostas~\cite{Karakostas}. We further show the surprising
fact that by strengthening the SDP with the (intractable) requirement that the
metric interpretation of the solution is an metric, we get an exact
relaxation (integrality gap is 1), and on the other hand if the solution is
arbitrarily close to being embeddable, the integrality gap may be as
big as . Finally, inspired by the above findings, we use ideas from the
integrality gap construction of Charikar \cite{Char02} to provide a family of
simple examples for negative type metrics that cannot be embedded into
with distortion better than 8/7-\eps. To this end we prove a new
isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof
of (current) Theorem
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