Anand Louis
ABSTRACT
Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: [ φ(S) def= (w(S,bar{S}))/(min set(w(S), w(bar(S)))) ≤ √(2 λ2) ] where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k ∈ [n], there exist ck disjoint subsets S1, ..., Sck, such that [ maxi φ(Si) ≤ C √(λk log k) ] where λk is the kth smallest eigenvalue of the normalized Laplacian and c<1,C>0 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 k, there is a subset S whose weight is at most a O(1/k) fraction of the total weight and φ(S) ≤ C √(λk log k). Both results are the best possible up to constant factors.
The underlying algorithmic problem, namely finding k 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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Index Terms
- Many sparse cuts via higher eigenvalues
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