Practical Minimum Cut Algorithms

@article{Henzinger2018PracticalMC,
  title={Practical Minimum Cut Algorithms},
  author={Monika Henzinger and Alexander Noe and Christian Schulz and Darren Strash},
  journal={Journal of Experimental Algorithmics (JEA)},
  year={2018},
  volume={23},
  pages={1 - 22}
}
The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi’s contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real… 

Shared-Memory Exact Minimum Cuts

TLDR
This paper engineer the fastest known exact algorithm for the minimum cut problem for an undirected edge-weighted graph by using a recently developed fast and parallel inexact minimum cut algorithm and using reductions that depend on this bound to reduce the size of the graph much faster than previously possible.

Practical Fully Dynamic Minimum Cut Algorithms

TLDR
The algorithm is the first implementation of a fully-dynamic algorithm that can maintain the global minimum cut of a graph with rapid update times and gives up to multiple orders of magnitude speedup compared to static approaches both on edge insertions and deletions.

A Simple Algorithm for Minimum Cuts in Near-Linear Time

TLDR
A self-contained version of Karger's algorithm is given with a new procedure, which produces a minimum cut on an m-edge, n-vertex graph in O(m \log^3 n) time with high probability, matching the complexity ofKarger's approach.

Improved Branching Strategies for Maximum Independent Sets

The NP-complete graph problem maximum independent set is that of finding a set of pairwise non adjacent vertices of largest cardinality [14]. Applications of this problem span multiple real-world

Engineering Nearly Linear-Time Algorithms for Small Vertex Connectivity

TLDR
According to experimental results on random graphs with planted vertex cuts, random hyperbolic graphs, and real world graphs with vertex connectivity between 4 and 15, the degree counting heuristic offers a factor of 2-4 speedup over the original non-degree counting version for most of the data.

Finding All Global Minimum Cuts In Practice

TLDR
A practically efficient algorithm that finds all global minimum cuts in huge undirected graphs using an optimized version of the algorithm of Nagamochi, Nakao and Ibaraki and a new linear time algorithm to find the most balanced minimum cuts.

In search of dense subgraphs: How good is greedy peeling?

TLDR
An efficient implementation of a greedy heuristic from the literature that is extremely fast and has some nice theoretical properties is provided, and a new heuristic algorithm is introduced that is built on top of the greedy and the exact methods.

Engineering Generalized Reductions for the Maximum Weight Independent Set Problem

TLDR
New data reduction techniques that are able to reduce previously irreducible graphs by use of generalized reduction rules are developed and an additional algorithm is developed that exploits the full potential of the struction by also allowing struction applications that blow-up the graph.

Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

TLDR
The two‐stage stochastic minimum s − t cut problem is introduced and it is proved that the considered problem is NP ‐hard in general, but admits a linear time solution algorithm when the graph is a tree.

Engineering Data Reduction for Nested Dissection

TLDR
This paper engineer new data reduction rules for the minimum fill-in problem, which significantly reduce the size of the graph while producing an equivalent (or near-equivalent) instance by applying both new and existing data reduction Rules exhaustively before nested dissection.

References

SHOWING 1-10 OF 49 REFERENCES

Communication-avoiding parallel minimum cuts and connected components

TLDR
Novel scalable parallel algorithms for finding global minimum cuts and connected components, which are important and fundamental problems in graph processing, and an approximate variant of the minimum cut algorithm, which approximates the exact solutions well while using a fractions of cores in a fraction of time are provided.

Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time

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

Local Flow Partitioning for Faster Edge Connectivity

TLDR
This flow subroutine is the first that is both local and produces low conductance cuts, and may be of independent interest.

A new approach to the minimum cut problem

TLDR
A randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability with a significant improvement over the previous time bounds based on maximum flows.

A simple min-cut algorithm

TLDR
An algorithm for finding the minimum cut of an undirected edge-weighted graph that has a short and compact description, is easy to implement, and has a surprisingly simple proof of correctness.

An efficient algorithm for the minimum capacity cut problem

TLDR
This work presents an alternative algorithm with the same worst-case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm.

A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems

An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that

Implementing an efficient minimum capacity cut algorithm

TLDR
An efficient implementation of theO(mn + n2 logn) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network and its running time is not significantly affected by the types of the networks being solved.

Minimum cuts in near-linear time

TLDR
A "semiduality" between minimum cuts and maximum spanning tree packings combined with the previously developed random sampling techniques is used and known time bounds for solving the minimum cut problem on undirected graphs are significantly improved.

A faster algorithm for finding the minimum cut in a graph

TLDR
This work considers the problem of finding the minimum capacity cut in a network G with n nodes, and shows how to reduce theRunning time of these 2n - 2 maximum flow algorithms to the running time for solving a single maximum flow problem.