8 research outputs found
Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces
We consider the problem of embedding a finite set of points x_1, ...x_n in R^d that satisfy l_2^2 triangle inequalities into l_1, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most sqrt{d}. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Pi such that the projections onto this subspace satisfy sum_{i,j in [n]} norm{Pi x_i - Pi x_j}_2^2 >= Omega(1) * sum_{i,j in [n]} norm{x_i - x_j}_2^2, then there is an embedding of the points into l_1 with O(sqrt{r}) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies lambda_r(G)/n >= Omega(1)*Phi_{SDP}(G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [Deshpande and Venkat, 2014], and [Deshpande, Harsha and Venkat 2016]
Approximation Algorithms for Semi-random Graph Partitioning Problems
In this paper, we propose and study a new semi-random model for graph
partitioning problems. We believe that it captures many properties of
real--world instances. The model is more flexible than the semi-random model of
Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and
Sipser.
We develop a general framework for solving semi-random instances and apply it
to several problems of interest. We present constant factor bi-criteria
approximation algorithms for semi-random instances of the Balanced Cut,
Multicut, Min Uncut, Sparsest Cut and Small Set Expansion problems. We also
show how to almost recover the optimal solution if the instance satisfies an
additional expanding condition. Our algorithms work in a wider range of
parameters than most algorithms for previously studied random and semi-random
models.
Additionally, we study a new planted algebraic expander model and develop
constant factor bi-criteria approximation algorithms for graph partitioning
problems in this model.Comment: To appear at the 44th ACM Symposium on Theory of Computing (STOC
2012
On Graph Crossing Number and Edge Planarization
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its
vertices into points of the plane, and its edges into continuous curves,
connecting the images of their endpoints. A crossing in such a drawing is a
point where two such curves intersect. In the Minimum Crossing Number problem,
the goal is to find a drawing of G with minimum number of crossings. The value
of the optimal solution, denoted by OPT, is called the graph's crossing number.
This is a very basic problem in topological graph theory, that has received a
significant amount of attention, but is still poorly understood
algorithmically. The best currently known efficient algorithm produces drawings
with crossings on bounded-degree graphs, while only a
constant factor hardness of approximation is known. A closely related problem
is Minimum Edge Planarization, in which the goal is to remove a
minimum-cardinality subset of edges from G, such that the remaining graph is
planar. Our main technical result establishes the following connection between
the two problems: if we are given a solution of cost k to the Minimum Edge
Planarization problem on graph G, then we can efficiently find a drawing of G
with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where is the maximum
degree in G. This result implies an O(n\cdot \poly(d)\cdot
\log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved
algorithms for special cases of the problem, such as, for example, k-apex and
bounded-genus graphs
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Combinatorial Optimization
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An Algorithm for the Graph Crossing Number Problem
We study the Minimum Crossing Number problem: given an -vertex graph ,
the goal is to find a drawing of in the plane with minimum number of edge
crossings. This is one of the central problems in topological graph theory,
that has been studied extensively over the past three decades. The first
non-trivial efficient algorithm for the problem, due to Leighton and Rao,
achieved an -approximation for bounded degree graphs. This
algorithm has since been improved by poly-logarithmic factors, with the best
current approximation ratio standing on O(n \poly(d) \log^{3/2}n) for graphs
with maximum degree . In contrast, only APX-hardness is known on the
negative side.
In this paper we present an efficient randomized algorithm to find a drawing
of any -vertex graph in the plane with O(OPT^{10}\cdot \poly(d \log
n)) crossings, where is the number of crossings in the optimal solution,
and is the maximum vertex degree in . This result implies an
\tilde{O}(n^{9/10} \poly(d))-approximation for Minimum Crossing Number, thus
breaking the long-standing -approximation barrier for
bounded-degree graphs
Bowdoin Orient v.137, no.1-25 (2007-2008)
https://digitalcommons.bowdoin.edu/bowdoinorient-2000s/1008/thumbnail.jp