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Approximate min–max relations for odd cycles in planar graphs

Samuel Fiorini, Nadia Hardy, Bruce Reed, Adrian Vetta
2006 Mathematical programming  
We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph.  ...  For the corresponding edge version of this problem, Král and Voss recently proved that this ratio is at most 2; we also give a short proof of their result.  ...  Acknowledgements We would like to thank both anonymous referees for providing a multitude of pertinent remarks, questions and suggestions which contributed to improve the paper.  ... 
doi:10.1007/s10107-006-0063-7 fatcat:uqpzp4fe55dvnon534gurx5tvy

Approximate Min-max Relations for Odd Cycles in Planar Graphs [chapter]

Samuel Fiorini, Nadia Hardy, Bruce Reed, Adrian Vetta
2005 Lecture Notes in Computer Science  
We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph.  ...  For the corresponding edge version of this problem, Král and Voss recently proved that this ratio is at most 2; we also give a short proof of their result.  ...  Acknowledgements We would like to thank both anonymous referees for providing a multitude of pertinent remarks, questions and suggestions which contributed to improve the paper.  ... 
doi:10.1007/11496915_4 fatcat:vx23og2cq5evbcgz6ewvctusdi

Approximating Unique Games Using Low Diameter Graph Decomposition [article]

Vedat Levi Alev, Lap Chi Lau
2017 arXiv   pre-print
A corollary is an improved approximation algorithm for the MaxCut problem for K_r-minor free graphs.  ...  We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition.  ...  We thank Tsz Chiu Kwok, Akshay Ramachandran and Hong Zhou for useful discussions.  ... 
arXiv:1702.06969v5 fatcat:f2gxofddxnduvavidrtsvk2zp4

Packing cycles in planar and bounded-genus graphs [article]

Niklas Schlomberg and Hanjo Thiele and Jens Vygen
2022 arXiv   pre-print
We also obtain approximate min-max theorems of the Erdős–Pósa type. For example, the minimum feedback vertex set in a planar digraph is at most 12 times the maximum number of vertex-disjoint cycles.  ...  For example, three families that satisfy the above properties are (i) all cycles in a directed or undirected graph, (ii) all odd cycles in an undirected graph, and (iii) all cycles in an undirected graph  ...  For planar graphs, our results complement previous work on the cycle transversal problem and thus yield new approximate min-max theorems.  ... 
arXiv:2207.00450v1 fatcat:mov37crprfet5pim3ilrdtkbw4

Properties of an Approximability-related Parameter on Circular Complete Graphs

Robert Engström, Tommy Färnqvist, Peter Jonsson, Johan Thapper
2009 Electronic Notes in Discrete Mathematics  
Färnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of Max H-Col.  ...  that the subgraph has a homomorphism to H; note that for H = K k this problem is equivalent to Max k-cut.  ...  [3] present concrete approximation ratios for certain graphs (such as the odd cycles) and near-optimal asymptotic results for large graph classes.  ... 
doi:10.1016/j.endm.2009.11.020 fatcat:fmsfdnafknda5kevwzg3bincae

Recent techniques and results on the Erdős–Pósa property

Jean-Florent Raymond, Dimitrios M. Thilikos
2017 Discrete Applied Mathematics  
Several min-max relations in graph theory can be expressed in the framework of the Erdős-Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs.  ...  We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.  ...  Fomin, Gwenaël Joret, Daniel Lokshtanov, Ignasi Sau, and Saket Saurabh for sharing their ideas and viewpoints on the Erdős-Pósa property, as well as the anonymous referees for their useful comments.  ... 
doi:10.1016/j.dam.2016.12.025 fatcat:rx6m6x6z6ja6zdjl7t2inj26ra

Space complexity: what makes planar graphs special?

Samir Datta, Raghav Kulkarni
2013 Bulletin of the European Association for Theoretical Computer Science  
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.  ...  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.  ...  For instance: does Max-Cut in planar graph have a Log-space approximation scheme ?  ... 
dblp:journals/eatcs/DattaK13 fatcat:errya5hbabeg5bh2qph5yobtiy

A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane [article]

Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri
2020 arXiv   pre-print
In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APXhard [Eiben and Kanj, TALG, 2020].  ...  This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known.  ...  a natural extension of the min-max relation for shortest paths, and that the methods for proving Theorem 1.4 led to a proof of Theorem 1.1 as well.  ... 
arXiv:2011.14463v1 fatcat:xgfjfn7vpnfdflstqs2ekvme6q

Spanning tree congestion of planar graphs

Hiu Law, Siu Leung, Mikhail Ostrovskii
2014 Involve. A Journal of Mathematics  
123 The paper is devoted to estimates of the spanning tree congestion for some planar graphs.  ...  The main results of the paper: (1) We almost determined (up to ±1) the maximal possible spanning tree congestion for planar graphs. (2) The value of congestion indicator introduced in Ostrovskii [Discrete  ...  For planar graphs the spanning tree congestion is closely related with the widely used notion of stretch, see [25, p. 166 ].  ... 
doi:10.2140/involve.2014.7.205 fatcat:pp46n6tcyjb43bxvmn4ntnhvku

An Approximation Algorithm for Fully Planar Edge-Disjoint Paths [article]

Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Kevin Schewior, Jens Vygen
2020 arXiv   pre-print
We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph.  ...  By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge.  ...  Hence theÕ(n 9/19 )-approximation algorithm for NDP with planar supply graphs [8] implies the same approximation ratio for EDP with G + H planar.  ... 
arXiv:2001.01715v1 fatcat:gsvqtvjpk5bvfmfrx3t62tjfby

Page 3934 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
For a simple graph containing no even subdivision of K4, these results imply that every rank facet is due either to an edge or to an odd cycle; consequently, the max- min relation specializes to give that  ...  an integral max-min relation: In a simple graph, the maximum size of a stable set is equal to the minimum (weighted) value of a cover of nodes by a-critical subgraphs.  ... 

Storage Capacity as an Information-Theoretic Vertex Cover and the Index Coding Rate [article]

Arya Mazumdar, Andrew McGregor, Sofya Vorotnikova
2018 arXiv   pre-print
Since the storage capacity is intimately related to the index coding rate, we get a 2 approximation of index coding rate for planar graphs and 3/2 approximation for triangle-free planar graphs.  ...  for triangle-free planar graphs.  ...  In section 4, we prove our approximation results for planar graphs that include a 3/2 approximation for Cap(G), 2 approximation for Ind(G) and a PTAS for Ind q (G).  ... 
arXiv:1706.09197v3 fatcat:g4lqqd6ym5hfzcnfugkbjp53zq

Recent techniques and results on the Erdős-Pósa property [article]

Jean-Florent Raymond, Dimitrios M. Thilikos
2016 arXiv   pre-print
Several min-max relations in graph theory can be expressed in the framework of the Erdős-Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs.  ...  We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.  ...  Fomin, Gwenaël Joret, Daniel Lokshtanov, Ignasi Sau, and Saket Saurabh for sharing their ideas and viewpoints on the Erdős-Pósa property, as well as the anonymous referees for their useful comments.  ... 
arXiv:1603.04615v4 fatcat:g7khz7utdfewzb24rpmbe45vke

Approximability Distance in the Space of H-Colourability Problems [chapter]

Tommy Färnqvist, Peter Jonsson, Johan Thapper
2009 Lecture Notes in Computer Science  
Specifically, the approximation algorithms for MAX CUT by Goemans and Williamson and MAX k-CUT by Frieze and Jerrum can be used to yield non-trivial approximation results for MAX H -COL.  ...  A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph.  ...  The girth of a graph is the length of a shortest cycle contained in the graph. Similarly, the odd girth of a graph gives the length of a shortest odd cycle in the graph.  ... 
doi:10.1007/978-3-642-03351-3_11 fatcat:lntdbuq5x5gqnhq6ako5etm44a

Packing Cycles and Cuts in Undirected Graphs [chapter]

Alberto Caprara, Alessandro Panconesi, Romeo Rizzi
2001 Lecture Notes in Computer Science  
Essentially the same approach achieves constant approximation for "dense" graphs. We show that both problems are N P-hard for planar graphs.  ...  We study the complexity and approximability of Cut Packing and Cycle Packing.  ...  We would like to thank Zoltan Szigeti and Maxim Sviridenko for helpful discussions and email exchanges.  ... 
doi:10.1007/3-540-44676-1_43 fatcat:jxgegse77je4lothz4xleey4uq
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