A Linear-time Algorithm for Sparsification of Unweighted Graphs

TL;DR

This paper presents an optimal linear-time algorithm for sparsifying unweighted graphs, producing a sparse approximation that preserves cut weights within a specified error, improving efficiency over previous methods.

Contribution

The paper introduces a new linear-time algorithm for unweighted graph sparsification that matches the best known size bounds and is optimal in time complexity.

Findings

Achieves $O(m)$ time complexity for sparsification.

Produces a sparsifier with $O(n \, \log n/\epsilon^2)$ edges.

Matches the best known size bounds for sparsifiers.

Abstract

Given an undirected graph $G$ and an error parameter $\epsilon > 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_{\epsilon}$ with the following property: the weight of every cut in $G_{\epsilon}$ is within a factor of $(1\pm \epsilon)$ of the weight of the corresponding cut in $G$. If $G$ is unweighted, an $O(m\log n)$-time algorithm for constructing $G_{\epsilon}$ with $O(n\log n/\epsilon^2)$ edges in expectation, and an $O(m)$-time algorithm for constructing $G_{\epsilon}$ 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 $G_{\epsilon}$ 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 $G_{\epsilon}$. Our algorithm is Monte-Carlo, i.e. it produces the correct output with high probability, as are all efficient graph sparsification algorithms.