Combinatorica
Faster mixing via average conductance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Mathematical aspects of mixing times in Markov chains
Foundations and Trends® in Theoretical Computer Science
How to Play Unique Games Using Embeddings
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
A combinatorial, primal-dual approach to semidefinite programs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Full regularization path for sparse principal component analysis
Proceedings of the 24th international conference on Machine learning
Rounding Parallel Repetitions of Unique Games
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Max cut and the smallest eigenvalue
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Algorithmic extensions of cheeger's inequality to higher eigenvalues and partitions
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Many sparse cuts via higher eigenvalues
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Hi-index | 0.00 |
The spectral profile of a graph is a natural generalization of the classical notion of its Rayleigh quotient. Roughly speaking, given a graph G, for each 0G(δ) minimizes the Rayleigh quotient (from the variational characterization) of the spectral gap of the Laplacian matrix of G over vectors with support at most δ over a suitable probability measure. Formally, the spectral profile ΛG of a graph G is a function ΛG : [0,1/2] - R defined as: ΛG(δ) def= minx∈ RVd(supp(x))≤ δ (∑gij (xi-xj)2)/(∑i di xi2) where gij is the weight of the edge (i,j) in the graph, di is the degree of vertex i, and d(\supp(x)) is the fraction of edges incident on vertices within the support of vector x. While the notion of the spectral profile has numerous applications in Markov chain, it is also is closely tied to its isoperimetric profile of a graph. Specifically, the spectral profile is a relaxation for the problem of approximating edge expansion of small sets in graphs. In this work, we obtain an efficient algorithm that yields a log(1/δ)-factor approximation for the value of ΛG(δ). By virtue of its connection to edge-expansion, we also obtain an algorithm for the problem of approximating edge expansion of small linear sized sets in a graph. This problem was recently shown to be intimately connected to the Unique Games Conjecture in [18]. Finally, we extend the techniques to obtain approximation algorithms with similar guarantees for restricted eigenvalue problems on diagonally dominant matrices.