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### The Karger-Stein Algorithm is Optimal for k-cut [article]

Anupam Gupta, Euiwoong Lee, Jason Li
2019 arXiv   pre-print
In this work, we resolve the problem for general graphs, by showing that for any fixed k ≥ 2, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least Ô(n^-k), where Ô(·) hides  ...  Algorithms due to Karger-Stein and Thorup showed how to find such a minimum k-cut in time approximately O(n^2k-2).  ...  Our Techniques Let us first recall the Karger-Stein algorithm: Algorithm 1 Karger-Stein Algorithm 1: procedure Karger-Stein(G = (V, E, w), k ∈ N) Compute a minimum k-cut of G 2: while |V | > k do 3: Sample  ...

### Communication-efficient parallel multiway and approximate minimum cut computation [chapter]

Friedhelm Meyer auf der Heide, Gabriel Terán Martinez
1998 Lecture Notes in Computer Science
For this setting we present improved BSP implementations of the algorithm of Karger and Stein.  ...  A nice e ect, beside the optimality, is that communication is e cient for a large spectrum of BSP-parameters. In the case of the minimal cut problem our results are close to optimal.  ...  These algorithms are based on the contraction algorithm of Karger and Stein 13] . The question whether an optimal BSP-implementation for the new algorithm of Karger exists has not been answered yet.  ...

### Parallel and fast sequential algorithms for undirected edge connectivity augmentation

András A. Benczúr
1999 Mathematical programming
We also present new efficient subroutines for finding the so-called extreme sets and the cactus representation of min-cuts required in our algorithms.  ...  It is a known non-trivial property of the edge connectivity augmentation problem that there is a sequence of edge sets E 1 , E 2 , . . . , such that i≤τ E i optmially increases the connectivity by τ, for  ...  Many people helped with clarifying various parts of this paper: Lisa Fleischer in the cactus algorithm part; David Karger (my co-author in [6] ) in the extreme sets algorithm part; Michel Goemans (my  ...

### Multicriteria Global Minimum Cuts [chapter]

Amitai Armon, Uri Zwick
2004 Lecture Notes in Computer Science
We show that the AND-version of the multicriteria global minimum cut problem is polynomial for any fixed number k of criteria.  ...  Given k bounds b1, b2, . . . , b k , the basic multicriteria decision problem is whether there exists a cut C of the graph that can be purchased using a budget of bi units of the i-th criterion, for 1  ...  More specifically, Karger and Stein [13] showed that for every α ≥ 1, not necessarily integral, the number of α-approximate solutions is only O(n 2α ).  ...

### Page 2777 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews
For this setting we present improved BSP implementations of the algorithm of Karger and Stein.  ...  Karger and Stein have presented a recursive contraction algorithm to solve minimum cut problems.  ...

### Scalable Algorithm for Higher-Order Co-Clustering via Random Sampling

Daisuke Hatano, Takuro Fukunaga, Takanori Maehara, Ken-ichi Kawarabayashi
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
Our algorithm is based on the random sampling technique, which has been successfully applied to graph cut problems.  ...  Each iteration of our algorithm runs in polynomial on the size of hypergraphs, and thus it performs well even for higher-order tensors, which are difficult to deal with for state-of-the-art algorithm.  ...  was extended by Karger and Stein (1996) for the k-way cut problem.  ...

### Counting small cuts in a graph [article]

Barbara Geissmann, Rastislav Šrámek
2013 arXiv   pre-print
We study the minimum cut problem in the presence of uncertainty and show how to apply a novel robust optimization approach, which aims to exploit the similarity in subsequent graph measurements or similar  ...  With experiments we show that the approach works well when compared to other approaches that are also oblivious towards the relationship between the input datasets.  ...  We will use the approach of Karger and Stein because it is the fastest currently known algorithm.  ...

### On cutting a few vertices from a graph

Uriel Feige, Robert Krauthgamer, Kobbi Nissim
2003 Discrete Applied Mathematics
For general k (i.e. k is part of the input and may depend on n) this problem is NP-hard. We present for this problem a randomized approximation algorithm, which is useful when k is relatively small.  ...  We consider the problem of ÿnding in an undirected graph a minimum cut that separates exactly a given number k of vertices.  ...  The algorithm Our algorithm for ÿnding a (k; n − k) cut (of nearly minimum cost) uses the random edge contraction technique of Karger and Stein [5] .  ...

### Practical Performance of Efficient Minimum Cut Algorithms

M. Jünger, G. Rinaldi, S. Thienel
2000 Algorithmica
We provide a brief overview of the most important algorithms for the minimum capacity cut problem and compare these methods both on problem instances from the literature and on problem instances originating  ...  from the solution of the traveling salesman problem by branch-and-cut. 3.  ...  Acknowledgements In an earlier version of this article, we had used a version of KS that required jV j 2 space as in the conference version of KS96 .  ...

### Minimum cuts in near-linear time

David R. Karger
2000 Journal of the ACM
We significantly improve known time bounds for solving the minimum cut problem on undirected graphs.  ...  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 ᏺᏯ parallelization.  ...  Thanks to Robert Tarjan for some helpful references and comments on dynamic merging. Thanks to Eric Lehman and Matt Levine for some careful reading and suggestions for presentation improvements.  ...

### Minimum cuts in near-linear time

David R. Karger
1996 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
We significantly improve known time bounds for solving the minimum cut problem on undirected graphs.  ...  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 ᏺᏯ parallelization.  ...  Thanks to Robert Tarjan for some helpful references and comments on dynamic merging. Thanks to Eric Lehman and Matt Levine for some careful reading and suggestions for presentation improvements.  ...

### Practical Minimum Cut Algorithms [chapter]

Monika Henzinger, Alexander Noe, Christian Schulz, Darren Strash
2018 2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)
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.  ...  Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art algorithms while  ...  Acknowledgements The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) /ERC grant agreement  ...

### LP Relaxation and Tree Packing for Minimum k-cuts [article]

Chandra Chekuri, Kent Quanrud, Chao Xu
2018 arXiv   pre-print
Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts.  ...  This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings.  ...  The randomized algorithm of Karger and Stein [15] runs inÕ(n 2(k−1) ) time and outputs the optimum cut with high probability.  ...

### A Simple Algorithm for Minimum Cuts in Near-Linear Time

Nalin Bhardwaj, Antonio J. Molina Lovett, Bryce Sandlund, Susanne Albers
2020 Scandinavian Workshop on Algorithm Theory
This procedure can be used in place of the complicated subroutine given in Karger's near-linear time minimum cut algorithm [Karger, 2000].  ...  We give a self-contained version of Karger's algorithm with the new procedure, which is easy to state and relatively simple to implement.  ...  The Karger-Stein algorithm achieves runtime O(n 2 log 3 n), finding the minimum cut with high probability.  ...

### LP Relaxation and Tree Packing for Minimum k-cuts

Chandra Chekuri, Kent Quanrud, Chao Xu, Michael Wagner
2018 ACM-SIAM Symposium on Discrete Algorithms
Karger used spanning tree packings [14] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts.  ...  Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [29] .  ...  The randomized algorithm of Karger and Stein [15] runs in Õ(n 2(k−1) ) time and outputs the optimum cut with high probability.  ...
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