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Bidimensionality and Parameterized Algorithms (Invited Talk)

Dimitrios M. Thilikos, Marc Herbstritt
2015 International Symposium on Parameterized and Exact Computation  
Under certain additional conditions, mainly of logical and combinatorial nature, such problems admit subexponential parameterized algorithms and linear kernels when their inputs are restricted to certain  ...  We provide an exposition of the main results of the theory of bidimensionality in parameterized algorithm design.  ...  Kolliopoulos, and Spyros Maniatis, for their helpful remarks on the manuscript.  ... 
doi:10.4230/lipics.ipec.2015.1 dblp:conf/iwpec/Thilikos15 fatcat:45a2txmf45bndj34diwdelb5na

Decomposition of Map Graphs with Applications

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
For Longest Cycle/Path, these are the first subexponential-time parameterized algorithm on map graphs.  ...  The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs.  ...  parameterized subexponential-time algorithms on planar graphs (see, e.g., [1, 2, 14, 27] ).  ... 
doi:10.4230/lipics.icalp.2019.60 dblp:conf/icalp/FominLP0Z19 fatcat:f5wd3acl4fbcbh5rfd6tvsdooi

Optimality Program in Segment and String Graphs [chapter]

Édouard Bonnet, Paweł Rzążewski
2018 Lecture Notes in Computer Science  
Intrigued by an algorithm packing curves in 2 O(n 2/3 log n) by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs)  ...  Planar graphs are known to allow subexponential algorithms running in time 2 O( for most of the paradigmatic problems, while the brute-force time 2 Θ(n) is very likely to be asymptotically best on general  ...  to generalize or extend the subexponential (parameterized) algorithms to classes of graphs which do not fall into those categories.  ... 
doi:10.1007/978-3-030-00256-5_7 fatcat:63oa6burhndrbnpfoamjctqvsu

Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

Fedor V. Fomin, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, Saket Saurabh
2016 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)  
for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound  ...  Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality.  ...  ACKNOWLEDGEMENTS The authors thank Felix Reidl, Fernando Sánchez Villaamil, Somnath Sikdar, and Yngve Villanger for some preliminary discussions on the topic of this paper, as well as an anynomous reviewer  ... 
doi:10.1109/focs.2016.62 dblp:conf/focs/FominLMPPS16 fatcat:dyyrhwi5yfgqrh2h2lnjn7pf2u

Optimality program in segment and string graphs [article]

Édouard Bonnet, Paweł Rzążewski
2018 arXiv   pre-print
Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs  ...  is very likely to be asymptotically best on general graphs.  ...  The second author is grateful to organizers of Homonolo 2016 and 2017, where part of the work was done, and also to Karolina Okrasa for inspiring discussions on the topic.  ... 
arXiv:1712.08907v3 fatcat:hrrbw622ova7hebjvirt6rmdwy

Optimality Program in Segment and String Graphs

Édouard Bonnet, Paweł Rzążewski
2019 Algorithmica  
Intrigued by an algorithm packing curves in 2 O(n 2/3 log n) by Fox and Pach (SODA'11), we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs)  ...  best on general graphs.  ...  A natural line of research is to generalize or extend the subexponential (parameterized) algorithms to classes of graphs which do not fall into those categories.  ... 
doi:10.1007/s00453-019-00568-7 fatcat:y6rwhbhpdfhuzp4rhg65qx3z7e

Clique-Based Separators for Geometric Intersection Graphs [article]

Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, Leonidas Theocharous
2021 arXiv   pre-print
Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^2/3log n).  ...  We extend this result in several directions, obtaining the following results. Map graphs admit a balanced clique-based separator of weight O(√(n)), which is tight in the worst case.  ...  In the case of map graphs, one can derive 2 O( √ n log n) algorithms for Maximum Independent Set and Feedback Vertex Set by using the trivial bound k ⩽ n in the subexponential parameterized algorithms  ... 
arXiv:2109.09874v1 fatcat:n73uq67vajcijnzw3keyr2pjmu

Maximum Clique in Disk-Like Intersection Graphs [article]

Édouard Bonnet, Nicolas Grelier, Tillmann Miltzow
2020 arXiv   pre-print
On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set.  ...  We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex  ...  For, say, disk graphs, do we receive the mere abstract graph or a list of the disks specified by their centers and radii?  ... 
arXiv:2003.02583v1 fatcat:e5apjiz7gzgt3cj5s3ujijfjda

Decomposition of Map Graphs with Applications [article]

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi
2019 arXiv   pre-print
For Longest Cycle/Path, these are the first subexponential-time parameterized algorithms on map graphs.  ...  The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs.  ...  For parameterized subexponential-time algorithms on map graphs, the situation is less explored.  ... 
arXiv:1903.01291v1 fatcat:vrl74ssdu5d5lgo5oo3dxzlyei

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, Tom C. van der Zanden
2018 Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2018  
Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information.  ...  CCS CONCEPTS • Theory of computation → Computational geometry; Parameterized complexity and exact algorithms; *  ...  They give a general framework to obtain subexponential algorithms on planar graphs or, more generally, on H -minor free graphs.  ... 
doi:10.1145/3188745.3188854 dblp:conf/stoc/BergBKMZ18 fatcat:on76caz7bfd5xdcgmuewaijeqq

Distributed minimum time servicing for a team of Dubins vehicles

Giuseppe Notarstefano, Paola Pedone
2008 2008 47th IEEE Conference on Decision and Control  
We show that this problem can be approximated by an abstract linear program, namely a generalized version of linear programming, that can be solved in a distributed way over a network.  ...  In [8] a distributed algorithm, called FloodBasis algorithm, was introduced to solve the network abstract linear program in a finite number of rounds under suitable assumptions on the communication graph  ...  Proposition 5 . 1 ( 51 Negative result for the disk graph): Let S be a robotic network of Dubins vehicles communicating according to the disk graph G disk = (I, E disk ).  ... 
doi:10.1109/cdc.2008.4739337 dblp:conf/cdc/NotarstefanoP08 fatcat:7yggpuhzczfyrmmlofxrsjo6wa

Acyclic Orders, Partition Schemes and CSPs: Unified Hardness Proofs and Improved Algorithms

Peter Jonsson, Victor Lagerkvist, George Osipov
2021 Artificial Intelligence  
However, some of these problems (including several important temporal problems), despite likely not being solvable in subexponential time, admit non-trivial improved exponential-time algorithm, and we  ...  This result explains, in a uniform way, many existing hardness results from the literature, and shows that it is impossible to obtain subexponential time algorithms unless the exponential-time hypothesis  ...  An immediate consequence of lower bounds of this form is that we can immediately rule out certain kinds of algorithms for CSP(B ∨= ), e.g. algorithms based on graph-decomposition and k-consistency, which  ... 
doi:10.1016/j.artint.2021.103505 fatcat:vil7zf66i5hmlhqmffubhdfzf4

The parameterized complexity of finding a 2-sphere in a simplicial complex [article]

Benjamin Burton and Sergio Cabello and Stefan Kratsch and William Pettersson
2018 arXiv   pre-print
The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^o(sqrt(k))) time algorithm.  ...  We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K.  ...  In particular, this rules out subexponential-time algorithms for 3-SAT and, by appropriate reductions, for a host of other problems.  ... 
arXiv:1802.07175v1 fatcat:3e2zlnzkovbopbzhdr5qprr3fy

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs [article]

Vincent Cohen-Addad, Éric Colin de Verdière, Daniel Marx, Arnaud de Mesmay
2021 arXiv   pre-print
A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk.  ...  In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for  ...  By a slight abuse of language, we will often identify an abstract graph with its embedding.  ... 
arXiv:1903.08603v3 fatcat:uasnm56bvje37mnm2ociswceqa

Computational topology of graphs on surfaces [article]

Éric Colin de Verdière
2017 arXiv   pre-print
curves and graphs on surfaces, and solving standard graph algorithm problems more efficiently in the case of surface-embedded graphs.  ...  Typical questions include representing surfaces and graphs embedded on them computationally, deciding whether a graph embeds on a surface, solving computational problems related to homotopy, optimizing  ...  Methods applicable to several algorithmic problems have also emerged, in many cases extending previous ones invented for planar graphs: 1.  ... 
arXiv:1702.05358v2 fatcat:6vlahbtjnbaxhg4yy2zotfrsym
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