Kernelization of Graph Hamiltonicity - Proper H-Graphs

Steven Chaplick; Fedor V. Fomin; Petr A. Golovach; Dusan Knop; Peter Zeman

doi:10.1007/978-3-030-24766-9_22

Kernelization of Graph Hamiltonicity - Proper H-Graphs

Steven Chaplick, Fedor V. Fomin, Petr A. Golovach, Dusan Knop^*, Peter Zeman

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

Abstract

We obtain new polynomial kernels and compression algorithms for PATH COVER and CYCLE COVER, the well-known generalizations of the classical HAMILTONIAN PATH and HAMILTONIAN CYCLE problems. Our choice of parameterization is strongly influenced by the work of Biro, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi) graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results.

- PATH COVER admits a kernel of size O(parallel to H parallel to(8)), that is, we design an algorithm that for an n-vertex graph G and an integer k >= 1, in time polynomial in n and parallel to H parallel to, outputs a graph G' of size O(parallel to H parallel to(8)) and k'

- CYCLE COVER admits a compression of size O(parallel to H parallel to(10)) into another problem, called PRIZE COLLECTING CYCLE COVER, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of PRIZE COLLECTING CYCLE COVER of size O(parallel to H parallel to(10)).

In all our algorithms we assume that a proper H-decomposition is given as a part of the input.

Original language	English
Title of host publication	Algorithms and Data Structures. WADS 2019
Editors	Z. Friggstad, JR Sack, M. Salavatipour
Publisher	Springer, Cham
Pages	296-310
Number of pages	15
ISBN (Electronic)	978-3-030-24766-9
ISBN (Print)	978-3-030-24765-2
DOIs	https://doi.org/10.1007/978-3-030-24766-9_22
Publication status	Published - 2019
Externally published	Yes

Publication series

Series	Lecture Notes in Computer Science
Volume	11646
ISSN	0302-9743

Keywords

CIRCUITS
Cycle Cover
INTERVAL-GRAPHS
Kernelization
LOG N) ALGORITHM
PATH
Path Cover
Proper H-graphs

Access to Document

10.1007/978-3-030-24766-9_22

1 Article

KERNELIZATION OF GRAPH HAMILTONICITY: Proper H-Graphs
Chaplick, S., Fomin, F., Golovach, P. A., Knop, D. & Zeman, P., 2021, In: Siam Journal on Discrete Mathematics. 35, 2, p. 840-892 53 p.
Research output: Contribution to journal › Article › Academic › peer-review

Open Access

Cite this

@inproceedings{3814e2f6052f492badfd5a58843cbb3a,

title = "Kernelization of Graph Hamiltonicity - Proper H-Graphs",

abstract = "We obtain new polynomial kernels and compression algorithms for PATH COVER and CYCLE COVER, the well-known generalizations of the classical HAMILTONIAN PATH and HAMILTONIAN CYCLE problems. Our choice of parameterization is strongly influenced by the work of Biro, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi) graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results.- PATH COVER admits a kernel of size O(parallel to H parallel to(8)), that is, we design an algorithm that for an n-vertex graph G and an integer k >= 1, in time polynomial in n and parallel to H parallel to, outputs a graph G' of size O(parallel to H parallel to(8)) and k'- CYCLE COVER admits a compression of size O(parallel to H parallel to(10)) into another problem, called PRIZE COLLECTING CYCLE COVER, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of PRIZE COLLECTING CYCLE COVER of size O(parallel to H parallel to(10)).In all our algorithms we assume that a proper H-decomposition is given as a part of the input.",

keywords = "CIRCUITS, Cycle Cover, INTERVAL-GRAPHS, Kernelization, LOG N) ALGORITHM, PATH, Path Cover, Proper H-graphs",

author = "Steven Chaplick and Fomin, {Fedor V.} and Golovach, {Petr A.} and Dusan Knop and Peter Zeman",

note = "DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.",

year = "2019",

doi = "10.1007/978-3-030-24766-9_22",

language = "English",

isbn = "978-3-030-24765-2",

series = "Lecture Notes in Computer Science",

publisher = "Springer, Cham",

pages = "296--310",

editor = "Z. Friggstad and JR Sack and M. Salavatipour",

booktitle = "Algorithms and Data Structures. WADS 2019",

address = "Switzerland",

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Kernelization of Graph Hamiltonicity - Proper H-Graphs. / Chaplick, Steven; Fomin, Fedor V.; Golovach, Petr A. et al.
Algorithms and Data Structures. WADS 2019. ed. / Z. Friggstad; JR Sack; M. Salavatipour. Springer, Cham, 2019. p. 296-310 (Lecture Notes in Computer Science, Vol. 11646).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

TY - GEN

T1 - Kernelization of Graph Hamiltonicity - Proper H-Graphs

AU - Chaplick, Steven

AU - Fomin, Fedor V.

AU - Golovach, Petr A.

AU - Knop, Dusan

AU - Zeman, Peter

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2019

Y1 - 2019

N2 - We obtain new polynomial kernels and compression algorithms for PATH COVER and CYCLE COVER, the well-known generalizations of the classical HAMILTONIAN PATH and HAMILTONIAN CYCLE problems. Our choice of parameterization is strongly influenced by the work of Biro, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi) graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results.- PATH COVER admits a kernel of size O(parallel to H parallel to(8)), that is, we design an algorithm that for an n-vertex graph G and an integer k >= 1, in time polynomial in n and parallel to H parallel to, outputs a graph G' of size O(parallel to H parallel to(8)) and k'- CYCLE COVER admits a compression of size O(parallel to H parallel to(10)) into another problem, called PRIZE COLLECTING CYCLE COVER, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of PRIZE COLLECTING CYCLE COVER of size O(parallel to H parallel to(10)).In all our algorithms we assume that a proper H-decomposition is given as a part of the input.

AB - We obtain new polynomial kernels and compression algorithms for PATH COVER and CYCLE COVER, the well-known generalizations of the classical HAMILTONIAN PATH and HAMILTONIAN CYCLE problems. Our choice of parameterization is strongly influenced by the work of Biro, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi) graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results.- PATH COVER admits a kernel of size O(parallel to H parallel to(8)), that is, we design an algorithm that for an n-vertex graph G and an integer k >= 1, in time polynomial in n and parallel to H parallel to, outputs a graph G' of size O(parallel to H parallel to(8)) and k'- CYCLE COVER admits a compression of size O(parallel to H parallel to(10)) into another problem, called PRIZE COLLECTING CYCLE COVER, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of PRIZE COLLECTING CYCLE COVER of size O(parallel to H parallel to(10)).In all our algorithms we assume that a proper H-decomposition is given as a part of the input.

KW - CIRCUITS

KW - Cycle Cover

KW - INTERVAL-GRAPHS

KW - Kernelization

KW - LOG N) ALGORITHM

KW - PATH

KW - Path Cover

KW - Proper H-graphs

U2 - 10.1007/978-3-030-24766-9_22

DO - 10.1007/978-3-030-24766-9_22

M3 - Conference article in proceeding

SN - 978-3-030-24765-2

T3 - Lecture Notes in Computer Science

SP - 296

EP - 310

BT - Algorithms and Data Structures. WADS 2019

A2 - Friggstad, Z.

A2 - Sack, JR

A2 - Salavatipour, M.

PB - Springer, Cham

ER -

Kernelization of Graph Hamiltonicity - Proper H-Graphs

Abstract

Publication series

Keywords

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Research output

KERNELIZATION OF GRAPH HAMILTONICITY: Proper H-Graphs

Cite this