Approximations for the isoperimetric and spectral profile of graphs and related parameters

with , . STOC 2010. pdf

abstract

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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 \(0< \delta < 1\), the spectral profile \(\Lambda_G(\delta)\) minimizes the Rayleigh quotient (from the variational characterization) of the spectral gap of the Laplacian matrix of \(G\) over vectors with support at most \(\delta\) over a suitable probability measure. Formally, the spectral profile \(\Lambda_G\) of a graph \(G\) is a function \(\Lambda_G : [0,\frac{1}{2}] \rightarrow \R\) defined as: \[ \Lambda_G(\delta) \defeq \min_{\substack{x\in \R^V\\d(\supp(x))\le \delta}} \frac{\sum_{ij} g_{ij} (x_i-x_j)^2}{\sum_i d_i x_i^2} \mper \] where \(g_{ij}\) is the weight of the edge \((i,j)\) in the graph, \(d_i\) 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 \(\log(1/\delta)\)-factor approximation for the value of \(\Lambda_G(\delta)\). 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 [Raghavendra-Steurer’10].

Finally, we extend the techniques to obtain approximation algorithms with similar guarantees for restricted eigenvalue problems on diagonally dominant matrices.

keywords

small-set expansion, approximation algorithms, semidefinite programming.