Deterministic Edge Connectivity in Near-Linear Time

TL;DR

This paper introduces a deterministic near-linear time algorithm for computing edge connectivity and minimum cuts in simple undirected graphs, improving over previous methods and eliminating randomness.

Contribution

It presents the first deterministic near-linear time algorithm for edge connectivity and minimum cut, using a novel contraction technique and low-conductance cuts without randomization.

Findings

First deterministic near-linear time algorithm for edge connectivity.

Constructs cactus representation of all minimum cuts efficiently.

Outperforms previous deterministic algorithms in time complexity.

Abstract

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts. The previous fastest deterministic algorithm by Gabow from STOC'91 took ~O(m+k^2 n), where k is the edge connectivity, but k could be Omega(n). At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes. Our main technical contribution is a near-linear time algorithm that contract vertex sets of a simple input graph G with minimum degree d, producing a multigraph with ~O(m/d) edges which preserves all minimum cuts of G with at least 2 vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.