In this paper, we introduce a variant of spectral sparsification, called
probabilistic (ε,δ)-spectral sparsification. Roughly speaking,
it preserves the cut value of any cut (S,Sc) 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(nlogn/ε2) edges in time O~(n/ε2δ)
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An O~(n/OPT+n3/2+t) time O(logn/t)-approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A n1+o(1)/ε4 time approximation minimum s-t cut algorithm
with an εn additive error