Probabilistic Spectral Sparsification In Sublinear Time

MIT Abstract. In this paper, we introduce a variant of spectral sparsification, called proba-bilistic (ε, δ)-spectral sparsification. Roughly speaking, it preserves the cut value of any cut (S, S c) with an 1 ± ε multiplicative error and a δ |S| additive error. We show how to produce a probabilistic (ε, δ)-spectral sparsifier with O(n log n/ε 2) edges in time˜O(n/ε 2 δ) time for unweighted undirected graph. This gives fastest known sub-linear time algorithms for different cut problems on unweighted undirected graph such as • Añ O(n/OP T + n 3/2+t) time O(log n/t)-approximation algorithm for the sparsest cut problem and the balanced separator problem. • A n 1+o(1) /ε 4 time approximation minimum s-t cut algorithm with an εn additive error.