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The Clique Problem in Intersection Graphs of Ellipses and Triangles

Christoph Ambühl, Uli Wagner
2005 Theory of Computing Systems  
Here, we consider the CLIQUE problem for intersection graphs of ellipses, which, in a sense, interpolate between disks and line segments, and show that the problem is APX -hard in that case.  ...  To our knowledge, this is the first hardness result for the CLIQUE problem in intersection graphs of convex objects with finite description complexity.  ...  Acknowledgments We thank Thomas Erlebach for introducing us to the CLIQUE problem in intersection graphs, and for many helpful comments and discussions.  ... 
doi:10.1007/s00224-005-1141-6 fatcat:hpicyjkvnjaldfakzdclh5r27a

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Pawel Rzazewski, Florian Sikora, Marc Herbstritt
2018 International Symposium on Computational Geometry  
In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks.  ...  A (unit) disk graph is the intersection graph of closed (unit) disks in the plane.  ...  APX-hardness was shown for Maximum Clique in the intersection graphs of (non-filled) ellipses and triangles by Ambühl and Wagner [4] .  ... 
doi:10.4230/lipics.socg.2018.12 dblp:conf/compgeom/BonnetG0RS18 fatcat:67msd4qudbasro5p2neguvcy5m

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs [article]

Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora
2018 arXiv   pre-print
In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks.  ...  A (unit) disk graph is the intersection graph of closed (unit) disks in the plane.  ...  APX-hardness was shown for Maximum Clique in the intersection graphs of (nonfilled) ellipses and triangles by Ambühl and Wagner [4] .  ... 
arXiv:1712.05010v2 fatcat:4s2pb7ozfrgtfgcdkcn627fnwe

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

Marthe Bonamy, Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, Florian Sikora, Stéphan Thomassé
2021 Journal of the ACM  
In stark contrast, Maximum Clique on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have  ...  A (unit) disk graph is the intersection graph of closed (unit) disks in the plane.  ...  APX-hardness was shown for Maximum Clique in the intersection graphs of (non-filled ) ellipses and triangles by Ambühl and Wagner [7] .  ... 
doi:10.1145/3433160 fatcat:2rka5djut5hmnjgufudj5dfgkm

Conservative edge sparsification for graph SLAM node removal

Nicholas Carlevaris-Bianco, Ryan M. Eustice
2014 2014 IEEE International Conference on Robotics and Automation (ICRA)  
This paper reports on optimization-based methods for producing a sparse, conservative approximation of the dense potentials induced by node marginalization in simultaneous localization and mapping (SLAM  ...  The proposed methods start with a sparse, but overconfident, Chow-Liu tree approximation of the marginalization potential and then use optimization-based methods to adjust the approximation so that it  ...  matrix as input as in [10] and [16] . • The computational complexity of our method is dependent only upon the size of the elimination clique, and not on the size of the graph beyond the clique.  ... 
doi:10.1109/icra.2014.6906954 dblp:conf/icra/Carlevaris-BiancoE14 fatcat:364fwmqjqjhuhbsrso6bwujc7i

Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique [article]

Valentin E. Brimkov, Konstanty Junosza-Szaniawski, Sean Kafer, Jan Kratochvíl, Martin Pergel, Paweł Rzążewski, Matthew Szczepankiewicz, Joshua Terhaar
2016 arXiv   pre-print
We study the Clique problem in classes of intersection graphs of convex sets in the plane.  ...  The problem is known to be NP-complete in convex-set intersection graphs and straight-line-segment intersection graphs, but solvable in polynomial time in intersection graphs of homothetic triangles.  ...  Acknowledgements We thank Michael Kaufmann for valuable discussions on the topic. Parts of the results have been presented at the following conferences: ISAAC 2012 [12] and IWCIA 2014 [6] .  ... 
arXiv:1411.2928v2 fatcat:qwscl37mznc3tc2poaw7cmhs4a

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2013 Discrete & Computational Geometry  
However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane.  ...  Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s)  ... 
doi:10.1007/s00454-013-9534-9 fatcat:sccdix27orhd5p2ucvcdfchhvq

Recognizing Objects Using Color-Annotated Adjacency Graphs [chapter]

Peter Tu, Richard Hartley, Tushar Saxena
1999 Lecture Notes in Computer Science  
Then the problem of recognizing the template object in the search image is reduced to the problem of approximately matching the template graph as a subgraph of the search image graph.  ...  The algorithm represents the objects in the template and search images by weighted adjacency graphs.  ...  the largest clique in the association graph.  ... 
doi:10.1007/3-540-46805-6_15 fatcat:3sp2rtluond5vjai6mkbw6jtuy

Worst-case comparison of valid inequalities for the TSP

Michel X. Goemans
1995 Mathematical programming  
The corresponding factor for the class of clique tree inequalities is 8, while it is 4 for the path configuration inequalities.  ...  We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst-case improvement resulting from their addition to the subtour polyhedron  ...  at least three, (5) the intersection graph of the handles and teeth is a tree.  ... 
doi:10.1007/bf01585563 fatcat:kfba666wmnaebjevfpezij36aa

The Clique Problem in Ray Intersection Graphs

Sergio Cabello, Jean Cardinal, Stefan Langerman
2013 Discrete & Computational Geometry  
The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard.  ...  Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph.  ...  GReGAS and ComPoSe) of the European Science Foundation.  ... 
doi:10.1007/s00454-013-9538-5 fatcat:od4miryyl5eapmri6noz3d6dai

Maximum Clique in Disk-Like Intersection Graphs [article]

Édouard Bonnet, Nicolas Grelier, Tillmann Miltzow
2020 arXiv   pre-print
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane.  ...  The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard.  ...  [9] for filled ellipses and filled triangles, and by Bonamy et al. [7] for ball graphs, and 4-dimensional unit ball graphs. Bonnet et al.  ... 
arXiv:2003.02583v1 fatcat:e5apjiz7gzgt3cj5s3ujijfjda

Closed trail distance in a biconnected graph

Vaclav Snasel, Pavla Drazdilova, Jan Platos, Baruch Barzel
2018 PLoS ONE  
Graphs describe and represent many complex structures in the field of social networks, biological, chemical, industrial and transport systems, and others.  ...  In this article, we propose a novel metric that takes into account the higher degree of connectivity on the part of the graph (for example, biconnected fullerene graphs and fulleroids).  ...  Acknowledgments We would like to thank to the Peter Schwerdtfeger, Lukas N. Wirz and James Avery for the providing of the fulleroid data and images to be used in our article.  ... 
doi:10.1371/journal.pone.0202181 pmid:30169516 pmcid:PMC6118363 fatcat:nandz7wav5dnnik3e2vucn2zyy

Hardness Results and Efficient Algorithms for Graph Powers [chapter]

Van Bang Le, Ngoc Tuy Nguyen
2010 Lecture Notes in Computer Science  
The k-th power H k of a graph H is obtained from H by adding new edges between every two distinct vertices having distance at most k in H.  ...  Lau and Corneil [Recognizing powers of proper interval, split and chordal graphs, SIAM J.  ...  In this and other figures, each ellipse corresponds to a clique and we omit the clique edges to keep the figures simpler.  ... 
doi:10.1007/978-3-642-11409-0_21 fatcat:eqwta4d65rg43pr6o4grlfyllq

Grouping Curved Lines

Paul L. Rosin
1994 Procedings of the British Machine Vision Conference 1994  
This paper examines the problem of automatically grouping image curves. In contrast, most previous work has been restricted to points and straight lines.  ...  Some of the computational aspects of the groupings of continuation, parallelism, and proximity are analysed, and the issues of neighbourhoods, combinatorix, and multiple scales are discussed.  ...  The biggest set of mutually parallel curves is given by the largest maximal clique in the graph.  ... 
doi:10.5244/c.8.26 dblp:conf/bmvc/Rosin94 fatcat:2bsvryutwfbrlby6ypkalq5fa4

The Complexity of Planar Counting Problems [article]

Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, Richard E. Stearns
1998 arXiv   pre-print
We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances.  ...  These problems include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.  ...  These suggestions significantly improved the quality of presentation and helped us in correcting a number of errors in the earlier draft.  ... 
arXiv:cs/9809017v1 fatcat:wsh2itr3eragldlkyso7ke2ivm
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