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max-cut and Containment Relations in Graphs [chapter]

Marcin Kamiński
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.  ... 
doi:10.1007/978-3-642-16926-7_4 fatcat:4c45evsppndzhe6sb2zyktsu3q

max-cut and containment relations in graphs

Marcin Kamiński
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.  ... 
doi:10.1016/j.tcs.2012.02.036 fatcat:5mfosv6qljg5llo47yrh7r46ee

Space complexity: what makes planar graphs special?

Samir Datta, Raghav Kulkarni
2013 Bulletin of the European Association for Theoretical Computer Science  
The design of efficient 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 ?  ... 
dblp:journals/eatcs/DattaK13 fatcat:errya5hbabeg5bh2qph5yobtiy

Improved algorithms for min cut and max flow in undirected planar graphs

Giuseppe F. Italiano, Yahav Nussbaum, Piotr Sankowski, Christian Wulff-Nilsen
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
Third, we present 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.  ... 
doi:10.1145/1993636.1993679 dblp:conf/stoc/ItalianoNSW11 fatcat:oktevvmzrjedfio6bg4gqaghpm

Partitioning planar graphs: a fast combinatorial approach for max-cut

F. Liers, G. Pardella
2010 Computational optimization and applications  
In this work, we present 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.  ... 
doi:10.1007/s10589-010-9335-5 fatcat:kechjjl52vexhdcrw2pvm76aw4

A fast algorithm for minimum weight odd circuits and cuts in planar graphs

Adam N. Letchford, Nicholas A. Pearson
2005 Operations Research Letters  
We give 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.  ... 
doi:10.1016/j.orl.2004.12.001 fatcat:yn4tv2xovvcodmbjdxiugofpwi

Maximum Cut Parameterized by Crossing Number [article]

Markus Chimani, Christine Dahn, Martina Juhnke-Kubitzke, Nils M. Kriege, Petra Mutzel, Alexander Nover
2019 arXiv   pre-print
Our 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.  ... 
arXiv:1903.06061v2 fatcat:cc7ynjtixrfehisgsxbmbrhapq

Constant factor approximation of vertex-cuts in planar graphs

Eyal Amir, Robert Krauthgamer, Satish Rao
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
We devise the first constant factor approximation 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.  ... 
doi:10.1145/780555.780557 fatcat:bbrufwdo3fg2rb4ajuxaxdqvja

Constant factor approximation of vertex-cuts in planar graphs

Eyal Amir, Robert Krauthgamer, Satish Rao
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
We devise the first constant factor approximation 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.  ... 
doi:10.1145/780542.780557 dblp:conf/stoc/AmirKR03 fatcat:y23imoi56rearnl4z57c7sd544

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]

Glencora Borradaile, Christian Wulff-Nilsen
2010 arXiv   pre-print
We give an O(n^1.5 n) time 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  ... 
arXiv:1008.4966v1 fatcat:43v35mijurhebcbmzbxzzaxccm

MAX-CUT and MAX-BISECTION are NP-hard on unit disk graphs

Josep Díaz, Marcin Kamiński
2007 Theoretical Computer Science  
We also show that λ-precision 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.  ... 
doi:10.1016/j.tcs.2007.02.013 fatcat:lepqj7lpsrhrvplgoezdbb23ki

Improved Minimum Cuts and Maximum Flows in Undirected Planar Graphs [article]

Giuseppe F. Italiano, Piotr Sankowski
2010 arXiv   pre-print
We remark that this is the first known non-trivial 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.  ... 
arXiv:1011.2843v2 fatcat:u6zohdu6bbam5by7tleainbham

Classification on the Computational Complexity of Spin Models [article]

Shi-Xin Zhang
2019 arXiv   pre-print
We conclude by 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.  ... 
arXiv:1911.04122v1 fatcat:b3sefrnk5ff2lbp7e2j2qpr2pm

Connected max cut is polynomial for graphs without K_5 e as a minor [article]

Brahim Chaourar
2019 arXiv   pre-print
CMAX 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] .  ... 
arXiv:1903.12641v2 fatcat:ywkhr25aynhoza4c2dlwqqxvze
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