Domination and Cut Problems on Chordal Graphs with Bounded Leafage

The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 201...

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Published in:	Algorithmica Vol. 86; no. 5; pp. 1428 - 1474
Main Authors:	Galby, Esther, Marx, Dániel, Schepper, Philipp, Sharma, Roohani, Tale, Prafullkumar
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.05.2024 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Chordal graphs Computer Science Computer Sciences Computer Systems Organization and Communication Networks Data Structures and Information Theory Datavetenskap (datalogi) Dominating set FPT algorithms Graphs Leafage Mathematics of Computing MultiCut with undeletable terminals Multiway cut with undeletable terminals Parameterization Theory of Computation Leafage MultiCut with undeletable terminals Multiway cut with undeletable terminals Dominating set Chordal graphs FPT algorithms
ISSN:	0178-4617, 1432-0541, 1432-0541
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Abstract	The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree . We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals . We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n O ( 1 ) -time.
AbstractList	The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time $$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree . We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals . We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple $$n^{\mathcal {O}(\ell )}$$ n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $$n^{{{\mathcal {O}}}(1)}$$ n O ( 1 ) -time. The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree . We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals . We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n O ( 1 ) -time. The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond[ESA2018, Algorithmica2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2O(ℓ2)·nO(1). We present a conceptually much simpler algorithm that runs in time 2O(ℓ)·nO(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple nO(ℓ)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in nO(1)-time. The leafage of a chordal graph G is the minimum integer such that G can be realized as an intersection graph of subtrees of a tree with leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time . We present a conceptually much simpler algorithm that runs in time . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in -time. The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2O(ℓ2)·nO(1). We present a conceptually much simpler algorithm that runs in time 2O(ℓ)·nO(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple nO(ℓ)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in nO(1)-time.
Author	Galby, Esther Sharma, Roohani Tale, Prafullkumar Marx, Dániel Schepper, Philipp
Author_xml	– sequence: 1 givenname: Esther surname: Galby fullname: Galby, Esther email: esther.galby@tuhh.de organization: Hamburg University of Technology – sequence: 2 givenname: Dániel surname: Marx fullname: Marx, Dániel organization: CISPA Helmholtz Center for Information Security – sequence: 3 givenname: Philipp surname: Schepper fullname: Schepper, Philipp organization: CISPA Helmholtz Center for Information Security – sequence: 4 givenname: Roohani surname: Sharma fullname: Sharma, Roohani organization: Max Planck Institute for Informatics, Saarland Informatics Campus – sequence: 5 givenname: Prafullkumar surname: Tale fullname: Tale, Prafullkumar organization: Indian Institute of Science Education and Research Pune
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Cites_doi	10.1137/20M1350571 10.1137/140961808 10.7151/dmgt.1061 10.1137/110855247 10.1007/s00453-021-00817-8 10.1137/12086217X 10.1007/978-3-319-21275-3 10.1007/s00453-020-00692-9 10.4153/CJM-1964-055-5 10.4064/fm-51-1-45-64 10.1007/s00453-022-00936-w 10.1007/s00453-016-0127-x 10.1016/j.ejor.2007.02.014 10.1016/j.tcs.2005.10.007 10.1016/0095-8956(74)90094-X 10.1016/0020-0190(85)90050-X 10.1016/0890-5401(87)90028-9 10.1137/140975279 10.1145/322123.322125 10.1016/j.dam.2012.12.006 10.1016/j.tcs.2013.01.009 10.1137/S0097539792238431 10.1002/net.3230150109 10.1145/2700209 10.1016/j.dam.2013.04.019 10.1016/j.dam.2012.03.021 10.1002/net.21975 10.1007/s00453-012-9731-6 10.1016/j.ejc.2011.09.031 10.1007/s00224-020-09967-8 10.1137/S0895480199359624 10.1016/j.dam.2021.08.034 10.1007/s00453-010-9411-3 10.1016/j.dam.2018.11.017 10.1016/0012-365X(74)90002-8 10.4153/CJM-1956-045-5 10.1016/0020-0190(84)90126-1 10.1016/j.tcs.2017.09.006 10.1007/978-3-642-04128-0_27 10.1007/978-3-540-30559-0_21 10.1007/978-3-030-96731-4_21 10.4230/LIPIcs.MFCS.2020.70 10.1007/3-540-57155-8_242 10.1007/978-3-031-06678-8_34 10.1007/978-3-031-15914-5_3 10.1137/1.9781611974331.ch77
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Keywords	Leafage MultiCut with undeletable terminals Multiway cut with undeletable terminals Dominating set Chordal graphs FPT algorithms
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Snippet	The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider... The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ... The leafage of a chordal graph G is the minimum integer ℓ such that G can be realized as an intersection graph of subtrees of a tree with ℓ leaves. We consider... The leafage of a chordal graph G is the minimum integer such that G can be realized as an intersection graph of subtrees of a tree with leaves. We consider...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Chordal graphs Computer Science Computer Sciences Computer Systems Organization and Communication Networks Data Structures and Information Theory Datavetenskap (datalogi) Dominating set FPT algorithms Graphs Leafage Mathematics of Computing MultiCut with undeletable terminals Multiway cut with undeletable terminals Parameterization Theory of Computation
Title	Domination and Cut Problems on Chordal Graphs with Bounded Leafage
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