Affiliation:
1. National Institute of Informatics, Chiyoda-ku, Tokyo, Japan
2. BARC, University of Copenhagen, Denmark
Abstract
We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is 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. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time.
The previous fastest deterministic algorithm by Gabow from STOC '91 took Õ(
m
+λ
2
n
), where λ is the edge connectivity, but λ can be as big as
n
−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability.
Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph
G
with minimum degree Δ, producing a multigraph Ḡ with Õ(
m
/Δ) edges, which preserves all minimum cuts of
G
with at least two 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.
Funder
VILLUM Foundation
JSPS KAKENHI
Danish Council for Independent Research
JST ERATO Kawarabayashi Large Graph Project
Basic Algorithms Research Copenhage
Investigator Grant
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Reference31 articles.
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3. Efim A. Dinitz A. V. Karzanov and Micael V. Lomonosov. 1976. A structure of the system of all minimum cuts of a graph. In Studies in Discrete Optimization A.A. Fridman (Ed.). 290--306. Efim A. Dinitz A. V. Karzanov and Micael V. Lomonosov. 1976. A structure of the system of all minimum cuts of a graph. In Studies in Discrete Optimization A.A. Fridman (Ed.). 290--306.
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