13,237 research outputs found
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
Incorporating Road Networks into Territory Design
Given a set of basic areas, the territory design problem asks to create a
predefined number of territories, each containing at least one basic area, such
that an objective function is optimized. Desired properties of territories
often include a reasonable balance, compact form, contiguity and small average
journey times which are usually encoded in the objective function or formulated
as constraints. We address the territory design problem by developing graph
theoretic models that also consider the underlying road network. The derived
graph models enable us to tackle the territory design problem by modifying
graph partitioning algorithms and mixed integer programming formulations so
that the objective of the planning problem is taken into account. We test and
compare the algorithms on several real world instances
Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap
Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
\phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut
Towards an SDP-based Approach to Spectral Methods: A Nearly-Linear-Time Algorithm for Graph Partitioning and Decomposition
In this paper, we consider the following graph partitioning problem: The
input is an undirected graph a balance parameter and
a target conductance value The output is a cut which, if
non-empty, is of conductance at most for some function
and which is either balanced or well correlated with all cuts of conductance at
most Spielman and Teng gave an -time
algorithm for and used it to decompose graphs
into a collection of near-expanders. We present a new spectral algorithm for
this problem which runs in time for
Our result yields the first nearly-linear time algorithm for the classic
Balanced Separator problem that achieves the asymptotically optimal
approximation guarantee for spectral methods. Our method has the advantage of
being conceptually simple and relies on a primal-dual semidefinite-programming
SDP approach. We first consider a natural SDP relaxation for the Balanced
Separator problem. While it is easy to obtain from this SDP a certificate of
the fact that the graph has no balanced cut of conductance less than
somewhat surprisingly, we can obtain a certificate for the stronger correlation
condition. This is achieved via a novel separation oracle for our SDP and by
appealing to Arora and Kale's framework to bound the running time. Our result
contains technical ingredients that may be of independent interest.Comment: To appear in SODA 201
On the Distributed Complexity of Large-Scale Graph Computations
Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where
machines jointly perform computations on graphs with nodes (typically, ). The input graph is assumed to be initially randomly partitioned among
the machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques
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