A Linear-time Algorithm for Sparsification of Unweighted Graphs

Given an undirected graph $G$ and an error parameter $ϵ>0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_{ϵ}$ with the following property: the weight of every cut in $G_{ϵ}$ is within a factor of $(1\pm ϵ)$ of the weight of the corresponding cut in $G$. If $G$ is unweighted, an $O(m\log n)$-time algorithm for constructing $G_{ϵ}$ with $O(n\log n\mathrm{/}{ϵ}^2)$ edges in expectation, and an $O(m)$-time algorithm for constructing $G_{ϵ}$ with $O(n\log^2 n/\epsilon^2)$ edges in expectation have recently been developed (Hariharan-Panigrahi, 2010). In this paper, we improve these results by giving an $O(m)$-time algorithm for constructing $G_{ϵ}$ with $O(n\log n/\epsilon^2)$ edges in expectation, for unweighted graphs. Our algorithm is optimal in terms of its time complexity; further, no efficient algorithm is known for constructing a sparser $G_{ϵ}$. Our algorithm is Monte-Carlo, i.e. it produces the correct output with high probability, as are all efficient graph sparsification algorithms.