2 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
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