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Local certification of graph decompositions and applications to minor-free classes [article]

Nicolas Bousquet, Laurent Feuilloley, Théo Pierron
2021 arXiv   pre-print
In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log  ...  In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢.  ...  Schmidt for pointing out a mistake in the characterization of 3-connexity we used in an earlier version of this paper. We also thank the reviewers of an earlier version for their comments.  ... 
arXiv:2108.00059v2 fatcat:pj4jdsicavgodc4emr4xdkprty

Brief Announcement: Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Nicolas Bousquet, Laurent Feuilloley, Théo Pierron, Seth Gilbert
2021
In this paper, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(  ...  In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢.  ...  This generalizes and simplifies the lower bounds of [7] which apply only to K k and K p,q -minor-free graphs, and use ad-hoc and more complicated techniques.  ... 
doi:10.4230/lipics.disc.2021.49 fatcat:3m2sdfgwirarbeq5bjc6twkn6y

Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Nicolas Bousquet, Laurent Feuilloley, Théo Pierron, Quentin Bramas, Vincent Gramoli, Alessia Milani
2022
In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log  ...  In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢.  ...  O P O D I S 2 0 2 Certification of Graph Decompositions and Applications to Minor-Free Classes 0 2 1 22:10 Local Certification of Graph Decompositions and Applications to Minor-Free Classes Lemma 6([  ... 
doi:10.4230/lipics.opodis.2021.22 fatcat:5st7tweif5dwheosl7a6cxrthe

Linearity of grid minors in treewidth with applications through bidimensionality

Erik D. Demaine, Mohammadtaghi Hajiaghayi
2008 Combinatorica  
Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors.  ...  We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor.  ...  Seymour for many helpful discussions and for providing a portal into the Graph Minor Theory and revealing some of its hidden structure that we use in this paper.  ... 
doi:10.1007/s00493-008-2140-4 fatcat:cff2tl55u5bubhmz65rewhxpoa

Recent Advances on the Graph Isomorphism Problem [article]

Martin Grohe, Daniel Neuen
2021 arXiv   pre-print
Our main focus will be on Babai's quasi-polynomial time isomorphism test and subsequent developments that led to the design of isomorphism algorithms with a quasi-polynomial parameterized running time  ...  We give an overview of recent advances on the graph isomorphism problem.  ...  In particular, neither of the decompositions applied for graphs of bounded genus and graphs of bounded tree width are suitable for graph classes that only exclude K h as a minor.  ... 
arXiv:2011.01366v2 fatcat:drrbl2cx3zabtkxo75f6nc5jlq

What can be certified compactly? [article]

Nicolas Bousquet and Laurent Feuilloley and Théo Pierron
2022 arXiv   pre-print
A certification with such small certificates is now called a compact local certification, and it has become the gold standard of the area, similarly to polynomial time for centralized computing.  ...  Local certification consists in assigning labels (called certificates) to the nodes of a network to certify a property of the network or the correctness of a data structure distributed on the network.  ...  For all , -minor-free graphs and -minor-free graphs can be certified with (log )-bit certificates.  ... 
arXiv:2202.06065v1 fatcat:6kg5qprkl5ai3dnpppehl6fdqe

A Meta-Theorem for Distributed Certification [article]

Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, Ioan Todinca
2021 arXiv   pre-print
Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean  ...  The verification of the certificates must be fast, and the certificates themselves must be small.  ...  The authors are thankful to Eric Remila for fruitful discussions on certification schemes related to the one considered in this paper.  ... 
arXiv:2112.03195v1 fatcat:vslmebivsbadxim52dsmwnkieq

Explicit Linear Kernels for Packing Problems

Valentin Garnero, Christophe Paul, Ignasi Sau, Dimitrios M. Thilikos
2018 Algorithmica  
that each of them contains a graph in F as a minor.  ...  Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.  ...  We would like to thank Archontia Giannopoulou for insightful discussions about the Erdős-Pósa property for scattered planar minors.  ... 
doi:10.1007/s00453-018-0495-5 fatcat:x6z74jcdxjcdrapn4xdsefnqvu

Explicit linear kernels for packing problems [article]

Valentin Garnero, Christophe Paul, Ignasi Sau, Dimitrios M. Thilikos
2016 arXiv   pre-print
that each of them contains a graph in F as a minor.  ...  Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.  ...  We would like to thank Archontia C. Giannopoulou for insightful discussions about the Erdős-Pósa property for scattered planar minors.  ... 
arXiv:1610.06131v1 fatcat:m2pxlgstgvf3vm6teqxfr3ehsu

What Can Be Certified Compactly? Compact local certification of MSO properties in tree-like graphs

Laurent Feuilloley, Nicolas Bousquet, Théo Pierron
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
Local certification consists in assigning labels (called certificates) to the nodes of a network to certify a property of the network or the correctness of a data structure distributed on the network.  ...  A certification with such small certificates is now called a compact local certification, and it has become the gold standard of the area, similarly to polynomial time for centralized computing.  ...  ACKNOWLEDGMENTS We thank Louis Esperet for fruitful discussions, Rotem Oshman for pointing out references about triangle-free graphs certification, Anca Muscholl for discussions about tree automata.  ... 
doi:10.1145/3519270.3538416 fatcat:4z2ials4ofhpblutphqq3o7mdm

Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs

Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Dimitrios M. Thilikos, Marc Herbstritt
2013 Symposium on Theoretical Aspects of Computer Science  
For Connected Dominating Set no polynomial kernel on H-topological-minor free graphs was known prior to our work.  ...  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an Htopological-minor free graph G on O(k) vertices such that G has  ...  Our results make use of the decomposition theorem of Grohe and Marx in [25] that, in turn, can be seen as an extension of seminal results of the Graph Minor Series by Robertson and Seymour [32].  ... 
doi:10.4230/lipics.stacs.2013.92 dblp:conf/stacs/FominLST13 fatcat:7xiyebshenbdrb7t6ihlnlldku

Narrowing the LOCALx2013CONGEST Gaps in Sparse Networks via Expander Decompositions [article]

Yi-Jun Chang, Hsin-Hao Su
2022 arXiv   pre-print
This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs.  ...  Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster  ...  In this paper, we generalize these results to show that all H-minor-free graphs admit an edge separator of size O( ∆|V |), so the approach discussed above is applicable to all H-minor-free graphs.  ... 
arXiv:2205.08093v1 fatcat:h2wxmhc3lnhtdcghxhogjqbgre

Creative heritage of Vyacheslav Sergeevich Tanaev: Seventieth anniversary

V. S. Gordon, M. Ya. Kovalyov, G. M. Levin, Yu. N. Sotskov, V. A. Strusevich, A. V. Tuzikov, Ya. M. Shafransky
2010 Automation and remote control  
His heritage is represented by numerous scientific works and the internationally acknowledged school of the scheduling theory and decomposition of the optimization problems that he founded. V.S.  ...  Tanaev, as well as his basic scientific findings in the scheduling theory, decomposition of the optimization problems, and other fields of the applied mathematics are outlined in brief.  ...  An assertion was proved that for any two generated circuit-free oriented graphsand ′′ it is possible to construct a sequence ′ , 1 , . . . , , ′′ of the generated circuit-free oriented graphs differing  ... 
doi:10.1134/s0005117910100024 fatcat:i3b27nhtoja25ngelzoj2m3bci

Catalan structures and dynamic programming in H-minor-free graphs

Frederic Dorn, Fedor V. Fomin, Dimitrios M. Thilikos
2012 Journal of computer and system sciences (Print)  
We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2 O ( √ k) · n O (1) steps.  ...  Our approach can be viewed as a general framework for obtaining time 2 O ( √ k) · n O (1) algorithms on H-minor-free graph classes.  ...  be extended to larger graph classes, such as graphs locally excluding some graph as a minor [7] or graphs of bounded expansion [30] .  ... 
doi:10.1016/j.jcss.2012.02.004 fatcat:y5cjuqys3rewxmccwfy2hrkuwe

Data-Compression for Parametrized Counting Problems on Sparse Graphs

Eun Jung Kim, Maria Serna, Dimitrios M. Thilikos, Michael Wagner
2018 International Symposium on Algorithms and Computation  
We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula φ with one free set variable to be interpreted as a vertex subset, we want to count  ...  In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free  ...  in minor-closed graph classes.  ... 
doi:10.4230/lipics.isaac.2018.20 dblp:conf/isaac/0002ST18 fatcat:xytw4ioxsbelxko3iv4tmutely
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