158 research outputs found

    Many Sparse Cuts via Higher Eigenvalues

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    Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset SS such that its expansion (a.k.a. conductance) is bounded as follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{\lambda_2} where ww is the total edge weight of a subset or a cut and Ξ»2\lambda_2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k∈[n]k \in [n], there exist ckck disjoint subsets S1,...,SckS_1, ..., S_{ck}, such that max⁑iΟ•(Si)≀CΞ»klog⁑k \max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k} where Ξ»i\lambda_i is the ithi^{th} smallest eigenvalue of the normalized Laplacian and c0c0 are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any kk, there is a subset SS whose weight is at most a \bigO(1/k) fraction of the total weight and Ο•(S)≀CΞ»klog⁑k\phi(S) \le C \sqrt{\lambda_k \log k}. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding kk subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right
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