On 09.04.2015 during our Foundation for Polish Science Algorithmic Miniworkshops we will host Mikkel Thorup from University of Copenhagen. He will give a talk on his recent result on “Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time”. The abstract is given below.

We plan to have a group discussion and joint pierogi lunch after his talk (sponsored by FNP).

Ken-ichi Kawarabayashi, National Institute of Informatics, Tokyo, Japan

Mikkel Thorup, University of Copenhagen, Denmark

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.