Faster parameterized algorithms for minor containment

The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H - Minor containment is one of the most important and technical parts...

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Vydané v:	Theoretical computer science Ročník 412; číslo 50; s. 7018 - 7028
Hlavní autori:	Adler, Isolde, Dorn, Frederic, Fomin, Fedor V., Sau, Ignasi, Thilikos, Dimitrios M.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Oxford Elsevier B.V 25.11.2011 Elsevier
Predmet:	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Branchwidth Combinatorial analysis Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Containment Derivation Discrete Mathematics Dynamic programming Exact sciences and technology Graph minor containment Graph minors Graph theory Graphs Graphs on surfaces Information retrieval. Graph Ingredients Mathematical models Mathematics Miscellaneous Networks Parameterized complexity Sciences and techniques of general use Theoretical computing Dynamic programming Branchwidth Graph minor containment Graph minors Parameterized complexity Graphs on surfaces Modification Edge(graph) Computer theory Decomposition method Algorithmics Graph theory Contraction Complexity Surface Polynomial time Packing Subgraph
ISSN:	0304-3975, 1879-2294
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Abstract	The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H - Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory. H - Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 1–9], providing an algorithm that in time O ( 3 k 2 ⋅ ( h + k − 1 ) ! ⋅ m ) decides if a graph G with m edges and branchwidth k , contains a fixed graph H on h vertices as a minor. In this work we improve the dependence on k of Hicks’ result by showing that checking if H is a minor of G can be done in time O ( 2 ( 2 k + 1 ) ⋅ log k ⋅ h 2 k ⋅ 2 2 h 2 ⋅ m ) . We set up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time 2 O ( k ) ⋅ h 2 k ⋅ 2 O ( h ) ⋅ n , with n = ∣ V ( G ) ∣ . Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment.
AbstractList	The theory of Graph Minors by Robertson and Seymour is one of the deepest and significant theories in modern Combinatorics. This theory has also a strong impact on the recent development of Algorithms, and several areas, like Parameterized Complexity, have roots in Graph Minors. Until very recently it was a common belief that Graph Minors Theory is mainly of theoretical importance. However, it appears that many deep results from Robertson and Seymour's theory can be also used in the design of practical algorithms. Minor containment testing is one of algorithmically most important and technical parts of the theory, and minor containment in graphs of bounded branchwidth is a basic ingredient of this algorithm. In order to implement minor containment testing on graphs of bounded branchwidth, Hicks [NETWORKS 04] described an algorithm, that in time $\mathcal{O}(3^{k^2}\cdot (h+k-1)!\cdot m)$ decides if a graph G with m edges and branchwidth k, contains a fixed graph H on h vertices as a minor. That algorithm follows the ideas introduced by Robertson and Seymour in [J'CTSB 95]. In this work we improve the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time $\mathcal{O}(2^{(2k +1 )\cdot \log k} \cdot h^{2k} \cdot 2^{2h^2} \cdot m)$. Our approach is based on a combinatorial object called rooted packing, which captures the properties of the potential models of subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first single-exponential algorithm for minor containment testing. Namely, it runs in time $2^{\mathcal{O}(k)} \cdot h^{2k} \cdot 2^{\mathcal{O}(h)} \cdot n$, with n=\|V(G)\|. Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction minor containment. The H -Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H -Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory. H -Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 1-9], providing an algorithm that in time O ( 3 k 2 a< ( h + k - 1 ) ! a< m ) decides if a graph G with m edges and branchwidth k , contains a fixed graph H on h vertices as a minor. In this work we improve the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time O ( 2 ( 2 k + 1 ) a< log k a< h 2 k a< 2 2 h 2 a< m ) . We set up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time 2 O ( k ) a< h 2 k a< 2 O ( h ) a< n , with n = a 4 V ( G ) a 4 . Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment. The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H - Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory. H - Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 1–9], providing an algorithm that in time O ( 3 k 2 ⋅ ( h + k − 1 ) ! ⋅ m ) decides if a graph G with m edges and branchwidth k , contains a fixed graph H on h vertices as a minor. In this work we improve the dependence on k of Hicks’ result by showing that checking if H is a minor of G can be done in time O ( 2 ( 2 k + 1 ) ⋅ log k ⋅ h 2 k ⋅ 2 2 h 2 ⋅ m ) . We set up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time 2 O ( k ) ⋅ h 2 k ⋅ 2 O ( h ) ⋅ n , with n = ∣ V ( G ) ∣ . Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment.
Author	Thilikos, Dimitrios M. Fomin, Fedor V. Sau, Ignasi Dorn, Frederic Adler, Isolde
Author_xml	– sequence: 1 givenname: Isolde surname: Adler fullname: Adler, Isolde email: iadler@informatik.uni-frankfurt.de organization: Institut für Informatik, Goethe-Universität, Frankfurt, Germany – sequence: 2 givenname: Frederic surname: Dorn fullname: Dorn, Frederic email: frederic.dorn@ii.uib.no organization: Department of Informatics, University of Bergen, Norway – sequence: 3 givenname: Fedor V. surname: Fomin fullname: Fomin, Fedor V. email: fedor.fomin@ii.uib.no organization: Department of Informatics, University of Bergen, Norway – sequence: 4 givenname: Ignasi surname: Sau fullname: Sau, Ignasi email: ignasi.sau@lirmm.fr organization: AlGCo project-team, CNRS, LIRMM, Montpellier, France – sequence: 5 givenname: Dimitrios M. surname: Thilikos fullname: Thilikos, Dimitrios M. email: sedthilk@math.uoa.gr organization: Department of Mathematics, National and Kapodistrian University of Athens, Greece
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Cites_doi	10.1145/1536414.1536476 10.1002/net.10099 10.7155/jgaa.00014 10.1137/1.9781611973082.60 10.1016/j.jctb.2011.07.004 10.1016/0012-365X(92)90687-B 10.1007/s00453-009-9296-1 10.1016/0196-6774(89)90006-0 10.1016/j.cosrev.2008.02.004 10.1007/BF01215352 10.1016/j.jctb.2004.08.001 10.1006/jctb.1995.1006 10.1145/1101821.1101823 10.1109/SFCS.2005.14 10.1016/j.jda.2010.02.002 10.1006/jctb.1994.1073 10.1145/44483.44491
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Snippet	The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after... The H -Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after... The theory of Graph Minors by Robertson and Seymour is one of the deepest and significant theories in modern Combinatorics. This theory has also a strong...
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SubjectTerms	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Branchwidth Combinatorial analysis Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Containment Derivation Discrete Mathematics Dynamic programming Exact sciences and technology Graph minor containment Graph minors Graph theory Graphs Graphs on surfaces Information retrieval. Graph Ingredients Mathematical models Mathematics Miscellaneous Networks Parameterized complexity Sciences and techniques of general use Theoretical computing
Title	Faster parameterized algorithms for minor containment
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