Minimum Cuts in Near-Linear Time

TL;DR

This paper introduces faster randomized algorithms for finding minimum cuts in undirected graphs, achieving near-linear time complexity and improving upon previous bounds with novel techniques.

Contribution

It presents a new semi-duality approach combined with random sampling to significantly improve minimum cut algorithms' efficiency.

Findings

Randomized algorithm finds a minimum cut in O(m log^3 n) time.

Simpler algorithm finds all minimum cuts in O(n^2 log n) time.

New bounds on the number of near minimum cuts and a space-efficient data structure.

Abstract

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner.