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

2019
*
arXiv
*
pre-print

In this work, we resolve

arXiv:1911.09165v1
fatcat:5mepusz5pbe63nzeyxlngwtf4m
*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]

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

1999
*
Mathematical programming
*

We also present new efficient subroutines

doi:10.1007/s101070050042
fatcat:aqig3b3eubffrf5eensolxjrxa
*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]

2004
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-540-30551-4_8
fatcat:eztrjyosbnaxha6nyr5zc5ijby
*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

2017
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

Our

doi:10.1609/aaai.v31i1.10914
fatcat:33s6w3gimbdsfoml66it2kve4y
*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]

2013
*
arXiv
*
pre-print

We study

arXiv:1304.7632v1
fatcat:serabego65bxzhcwub6xpibkaa
*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

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

2000
*
Algorithmica
*

We provide a brief overview of

doi:10.1007/s004539910009
fatcat:3hzyvmwpovaqxcrtmciashue7a
*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

2000
*
Journal of the ACM
*

We significantly improve known time bounds

doi:10.1145/331605.331608
fatcat:wndnu7xz7rfuxgg3figfxsiinm
*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

1996
*
Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
*

We significantly improve known time bounds

doi:10.1145/237814.237829
dblp:conf/stoc/Karger96
fatcat:dhlpiuorqbcpjgbycp7jobbim4
*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]

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]

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

2020
*
Scandinavian Workshop on Algorithm Theory
*

This procedure can be used in place of

doi:10.4230/lipics.swat.2020.12
dblp:conf/swat/BhardwajLS20
fatcat:eagcf4lvkbebfhu4nijzfedq4e
*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

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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