Myhill–Nerode Methods for Hypergraphs

Bevern, René van; Downey, Rodney G.; Fellows, Michael R.; Gaspers, Serge; Rosamond, Frances A.

doi:10.1007/s00453-015-9977-x

“…1-Fold 2-Interval Cover is NP-hard even if each input 2-interval contains at most three positions [3,4]. However, if each 2-interval contains at most two positions, then the problem is polynomialtime solvable by techniques from matching theory [3,4,10].…”

Section: C-interval Multicovermentioning

confidence: 98%

“…However, if each 2-interval contains at most two positions, then the problem is polynomialtime solvable by techniques from matching theory [3,4,10]. 1-Fold 3-Interval Cover is APX-hard due to a simple reduction from Set Cover [3,4,10].…”

Section: C-interval Multicovermentioning

confidence: 99%

See 1 more Smart Citation

Approximability and parameterized complexity of multicover by c-intervals

Bevern

¹

,

Hüffner

²

,

Kratsch

³

et al. 2015

Information Processing Letters

Self Cite

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“…The formalization given in this section has been known in the graph grammar literature from eighty's. We refer to [7,37] for a review on these topics. We would also like to mention that the materials presented in this subsection is not much more than a formalization of what is commonly known as nice tree decomposition [26].…”

Section: Construction Termsmentioning

confidence: 99%

Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded Treewidth

Lokshtanov¹,

Pilipczuk²,

Pilipczuk³

et al. 2017

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Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.Publisher's statement: "© 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works." A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe give a fixed-parameter tractable algorithm that, given a parameter k and two graphs G 1 , G 2 , either concludes that one of these graphs has treewidth at least k, or determines whether G 1 and G 2 are isomorphic. The running time of the algorithm on an n-vertex graph is 2 O(k 5 log k) · n 5 , and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth.Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2 O(k 5 log k) · n 5 time that, for a given graph G on n vertices, either concludes that the treewidth of G is at least k, or:• finds in an isomorphic-invariant way a graph c(G) that is isomorphic to G;• finds an isomorphism-invariant construction term -an algebraic expression that encodes G together with a tree decomposition of G of width O(k 4 ).Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for G 1 and G 2 are equal. * A preliminary version of this paper has been presented at FOCS 2014. D.

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“…We refer to [7,37] for a review on these topics. We would also like to mention that the materials presented in this subsection is not much more than a formalization of what is commonly known as nice tree decomposition [26].…”

Section: Construction Termsmentioning

confidence: 99%

Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded Treewidth

Lokshtanov

¹

,

Pilipczuk

²

,

Pilipczuk

³

et al. 2014

2014 IEEE 55th Annual Symposium on Foundations of Computer Science

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Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.Publisher's statement: "© 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works." A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe give a fixed-parameter tractable algorithm that, given a parameter k and two graphs G 1 , G 2 , either concludes that one of these graphs has treewidth at least k, or determines whether G 1 and G 2 are isomorphic. The running time of the algorithm on an n-vertex graph is 2 O(k 5 log k) · n 5 , and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth.Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2 O(k 5 log k) · n 5 time that, for a given graph G on n vertices, either concludes that the treewidth of G is at least k, or:• finds in an isomorphic-invariant way a graph c(G) that is isomorphic to G;• finds an isomorphism-invariant construction term -an algebraic expression that encodes G together with a tree decomposition of G of width O(k 4 ).Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for G 1 and G 2 are equal. * A preliminary version of this paper has been presented at FOCS 2014. D.

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Myhill–Nerode Methods for Hypergraphs

Cited by 12 publications

References 41 publications

Approximability and parameterized complexity of multicover by c-intervals

Approximability and parameterized complexity of multicover by c-intervals

Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded Treewidth

Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded Treewidth

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