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Approximate min–max relations for odd cycles in planar graphs

2006
*
Mathematical programming
*

We study the ratio between the minimum size of an

doi:10.1007/s10107-006-0063-7
fatcat:uqpzp4fe55dvnon534gurx5tvy
*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. ...##
###
Approximate Min-max Relations for Odd Cycles in Planar Graphs
[chapter]

2005
*
Lecture Notes in Computer Science
*

We study the ratio between the minimum size of an

doi:10.1007/11496915_4
fatcat:vx23og2cq5evbcgz6ewvctusdi
*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. ...##
###
Approximating Unique Games Using Low Diameter Graph Decomposition
[article]

2017
*
arXiv
*
pre-print

A corollary is an improved

arXiv:1702.06969v5
fatcat:f2gxofddxnduvavidrtsvk2zp4
*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. ...##
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Packing cycles in planar and bounded-genus graphs
[article]

2022
*
arXiv
*
pre-print

We also obtain

arXiv:2207.00450v1
fatcat:mov37crprfet5pim3ilrdtkbw4
*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. ...##
###
Properties of an Approximability-related Parameter on Circular Complete Graphs

2009
*
Electronic Notes in Discrete Mathematics
*

Färnqvist et al. have introduced a parameter on the space of

doi:10.1016/j.endm.2009.11.020
fatcat:fmsfdnafknda5kevwzg3bincae
*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. ...##
###
Recent techniques and results on the Erdős–Pósa property

2017
*
Discrete Applied Mathematics
*

Several

doi:10.1016/j.dam.2016.12.025
fatcat:rx6m6x6z6ja6zdjl7t2inj26ra
*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. ...##
###
Space complexity: what makes planar graphs special?

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

There are several new efficient algorithms

dblp:journals/eatcs/DattaK13
fatcat:errya5hbabeg5bh2qph5yobtiy
*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 ? ...##
###
A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane
[article]

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. ...

##
###
Spanning tree congestion of planar graphs

2014
*
Involve. A Journal of Mathematics
*

123 The paper is devoted to estimates of the spanning tree congestion

doi:10.2140/involve.2014.7.205
fatcat:pp46n6tcyjb43bxvmn4ntnhvku
*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 ]. ...##
###
An Approximation Algorithm for Fully Planar Edge-Disjoint Paths
[article]

2020
*
arXiv
*
pre-print

We devise a constant-factor

arXiv:2001.01715v1
fatcat:gsvqtvjpk5bvfmfrx3t62tjfby
*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*. ...##
###
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]

2018
*
arXiv
*
pre-print

Since the storage capacity is intimately

arXiv:1706.09197v3
fatcat:g4lqqd6ym5hfzcnfugkbjp53zq
*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). ...##
###
Recent techniques and results on the Erdős-Pósa property
[article]

2016
*
arXiv
*
pre-print

Several

arXiv:1603.04615v4
fatcat:g7khz7utdfewzb24rpmbe45vke
*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. ...##
###
Approximability Distance in the Space of H-Colourability Problems
[chapter]

2009
*
Lecture Notes in Computer Science
*

Specifically, the

doi:10.1007/978-3-642-03351-3_11
fatcat:lntdbuq5x5gqnhq6ako5etm44a
*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*. ...##
###
Packing Cycles and Cuts in Undirected Graphs
[chapter]

2001
*
Lecture Notes in Computer Science
*

Essentially the same approach achieves constant

doi:10.1007/3-540-44676-1_43
fatcat:jxgegse77je4lothz4xleey4uq
*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. ...
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