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max-cut and Containment Relations in Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

*For*the minor order, we show how to solve

*max*-

*cut*in polynomial time

*for*the class obtained by forbidding

*a*

*graph*with crossing number at most one (this generalizes

*a*known result

*for*K 5 -minor-free

*graphs*... We study

*max*-

*cut*in classes of

*graphs*defined by forbidding

*a*single

*graph*as

*a*subgraph, induced subgraph, or minor.

*For*the first two containment relations, we prove dichotomy theorems. ... An early result that is of interest to us is

*a*polynomial-time

*algorithm*

*for*

*max*-

*cut*in

*planar*

*graphs*. ...

##
###
max-cut and containment relations in graphs

2012
*
Theoretical Computer Science
*

*For*the minor order, we show how to solve

*max*-

*cut*in polynomial time

*for*the class obtained by forbidding

*a*

*graph*with

*a*single crossing (this generalizes

*a*known result

*for*K 5 -minor-free

*graphs*) and ... We study

*max*-

*cut*in classes of

*graphs*defined by forbidding finitely many

*graphs*as subgraphs, or

*a*single

*graph*as an induced subgraph or

*a*minor. ... Acknowledgements I am grateful to Vadim Lozin and the anonymous referees

*for*comments and suggestions that helped to improve the paper. ...

##
###
Space complexity: what makes planar graphs special?

2013
*
Bulletin of the European Association for Theoretical Computer Science
*

The design of efficient

dblp:journals/eatcs/DattaK13
fatcat:errya5hbabeg5bh2qph5yobtiy
*algorithms**for**planar**graphs*, as*a*field of research, is over forty year old and continues to be an exciting area. ... There are several new efficient*algorithms**for**a*variety of*graph*optimization problems that exploit*planarity*and, in general, the structure of low genus*graphs*and*graphs*with excluded minors. ...*For*instance: does*Max*-*Cut*in*planar**graph*have*a*Log-space approximation scheme ? ...##
###
Improved algorithms for min cut and max flow in undirected planar graphs

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

Third, we present

doi:10.1145/1993636.1993679
dblp:conf/stoc/ItalianoNSW11
fatcat:oktevvmzrjedfio6bg4gqaghpm
*a*fully dynamic*algorithm*maintaining the value of the min st-*cuts*and the*max*st-flows in an undirected plane*graph*(i.e.,*a**planar**graph*with*a*fixed embedding): our*algorithm*is able ... Second, we show how to achieve the same bound*for*the problem of computing*a**max*st-flow in an undirected*planar**graph*. ...*algorithm*to compute*a*min st-*cut**for*undirected*planar**graphs*. ...##
###
Partitioning planar graphs: a fast combinatorial approach for max-cut

2010
*
Computational optimization and applications
*

In this work, we present

doi:10.1007/s10589-010-9335-5
fatcat:kechjjl52vexhdcrw2pvm76aw4
*a*new and*simple**algorithm**for*determining maximum*cuts**for*arbitrary weighted*planar**graphs*. ... Whereas*for*general instances the*max*-*cut*problem is NPhard, it is polynomially solvable*for*certain classes of*graphs*. ... We are grateful to the referees*for*their valuable suggestions. In particular, we thank*a*referee who has suggested the*algorithmic*variant*for*the method of [45] outlined in the end of Section 5. ...##
###
A fast algorithm for minimum weight odd circuits and cuts in planar graphs

2005
*
Operations Research Letters
*

We give

doi:10.1016/j.orl.2004.12.001
fatcat:yn4tv2xovvcodmbjdxiugofpwi
*a**simple*O(n 3/2 log n)*algorithm**for*finding*a*minimum weight odd circuit in*planar**graphs*. By geometric duality, the same*algorithm*can be used to find minimum weight odd*cuts*. ...*For*general sparse*graphs*, the fastest known*algorithms**for*these two problems take O(n 2 log n) time and O(n 3 log n) time, respectively. ... We will give*a**simple*O(n 3/2 log n)*algorithm**for*both problems in the*planar*case. ...##
###
Maximum Cut Parameterized by Crossing Number
[article]

2019
*
arXiv
*
pre-print

Our

arXiv:1903.06061v2
fatcat:cc7ynjtixrfehisgsxbmbrhapq
*algorithm*achieves*a*running time of O(2^k p(n + k)), where p is the polynomial running time*for**planar**Max*-*Cut*. ... Given an edge-weighted*graph*G on n nodes, the NP-hard*Max*-*Cut*problem asks*for**a*node bipartition such that the sum of edge weights joining the different partitions is maximized. ... We end up with*a*set of up to 2 k*planar**graphs*, each of which can be solved using*a*known polynomial-time*Max*-*Cut**algorithm**for**planar**graphs*. ...##
###
Constant factor approximation of vertex-cuts in planar graphs

2003
*
Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
*

We devise the first constant factor approximation

doi:10.1145/780555.780557
fatcat:bbrufwdo3fg2rb4ajuxaxdqvja
*algorithm**for*minimum quotient vertex-*cuts*in*planar**graphs*. ... First, we prove*a*structural theorem*for**planar**graphs*, showing the existence of*a*nearoptimal quotient vertex-*cut*whose high-level structure is that of*a*bounded-depth tree. ... The*algorithms*proceed by searching*for**simple*cycles in the dual of the*planar**graph*. ...##
###
Constant factor approximation of vertex-cuts in planar graphs

2003
*
Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
*

We devise the first constant factor approximation

doi:10.1145/780542.780557
dblp:conf/stoc/AmirKR03
fatcat:y23imoi56rearnl4z57c7sd544
*algorithm**for*minimum quotient vertex-*cuts*in*planar**graphs*. ... First, we prove*a*structural theorem*for**planar**graphs*, showing the existence of*a*nearoptimal quotient vertex-*cut*whose high-level structure is that of*a*bounded-depth tree. ... The*algorithms*proceed by searching*for**simple*cycles in the dual of the*planar**graph*. ...##
###
Page 34 of Mathematical Reviews Vol. , Issue 91H
[page]

1991
*
Mathematical Reviews
*

The authors establish

*a*generalization of the well- known duality*for**planar**graphs*between minimal*cuts*and*simple*cycles to one involving*cuts*and so-called even-degree sets (only an even number of whose ... logn) time*algorithm**for*the problem of finding*a**max*- imum weight matching of*a**planar**graph*with real weights is presented, improving the previous individual bounds O(n?) [resp., O(n? ...##
###
Multiple source, single sink maximum flow in a planar graph
[article]

2010
*
arXiv
*
pre-print

We give an O(n^1.5 n) time

arXiv:1008.4966v1
fatcat:43v35mijurhebcbmzbxzzaxccm
*algorithm**for*finding the maximum flow in*a*directed*planar**graph*with multiple sources and*a*single sink. ... The techniques generalize to*a*subquadratic time*algorithm**for*bounded genus*graphs*. ... Using the maximum flow*algorithms**for*general*graphs*,*a*multiple-source, single (or multiple) sink*max*flow in directed*planar**graphs*can be solved in O(n 2 log n) time using Goldberg and Tarjan's preflow ...##
###
MAX-CUT and MAX-BISECTION are NP-hard on unit disk graphs

2007
*
Theoretical Computer Science
*

We also show that λ-precision

doi:10.1016/j.tcs.2007.02.013
fatcat:lepqj7lpsrhrvplgoezdbb23ki
*graphs*are*planar**for*λ > 1/ √ 2 and give*a*dichotomy theorem*for**max*-*cut*computational complexity on λ-precision unit disk*graphs*. ... We prove that the*max*-*cut*and*max*-bisection problems are NP-hard on unit disk*graphs*. ...*A*maximum*cut*is*a**cut*with maximum size and the*algorithmic**max*-*cut*problem consists in finding one given the input*graph*. ...##
###
Improved Minimum Cuts and Maximum Flows in Undirected Planar Graphs
[article]

2010
*
arXiv
*
pre-print

We remark that this is the first known non-trivial

arXiv:1011.2843v2
fatcat:u6zohdu6bbam5by7tleainbham
*algorithm**for*min-*cut*and*max*-flow problems in*a*dynamic setting. ... Third, we present*a*fully dynamic*algorithm*that is able to maintain information about minimum*cuts*and maximum flows in*a*plane*graph*(i.e.,*a**planar**graph*with*a*fixed embedding): our*algorithm*is able ... s, t)-*planar**algorithm*: this yields an O(n log 2 n) time to compute*a*minimum*cut**for*undirected*planar**graphs*. ...##
###
Classification on the Computational Complexity of Spin Models
[article]

2019
*
arXiv
*
pre-print

We conclude by

arXiv:1911.04122v1
fatcat:b3sefrnk5ff2lbp7e2j2qpr2pm
*a*brief discussion on the picture when quantum computation and quantum complexity theory are included. ... In this note, we provide*a*unifying framework to investigate the computational complexity of classical spin models and give the full classification on spin models in terms of system dimensions, randomness ... Acknowledgment: We thank Hong Yao and Zi-Xiang Li*for*useful discussions. ...##
###
Connected max cut is polynomial for graphs without K_5 e as a minor
[article]

2019
*
arXiv
*
pre-print

CMAX

arXiv:1903.12641v2
fatcat:ywkhr25aynhoza4c2dlwqqxvze
*CUT*is NP-hard even*for**planar**graphs*. In this paper, we prove that CMAX*CUT*is polynomial*for**graphs*without K_5 e as*a*minor. ... We deduce*a*quadratic time*algorithm**for*the minimum*cut*problem in the same class of*graphs*without computing the maximum flow. ... CMAX*CUT*has been proved NP-hard*for**planar**graphs*[12] and*a*linear time*algorithm**for*series parallel*graphs*is presented in [7] . ...
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