Cutwidth: Obstructions and Algorithmic Aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersion...

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Published in:	Algorithmica Vol. 81; no. 2; pp. 557 - 588
Main Authors:	Giannopoulou, Archontia C., Pilipczuk, Michał, Raymond, Jean-Florent, Thilikos, Dimitrios M., Wrochna, Marcin
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2019 Springer Nature B.V Springer Verlag
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Apexes Combinatorial analysis Computer Science Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Design parameters Discrete Mathematics Graph theory Graphs Layouts Mathematics of Computing Obstructions Submerging Theory of Computation Immersions Obstructions Cutwidth Fixed-parameter tractability cutwidth
ISSN:	0178-4617, 1432-0541
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Abstract	Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k . We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2 O ( k 3 log k ) . As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2 O ( k 2 log k ) · n , where k is the optimum width and n is the number of vertices. While being slower by a log k -factor in the exponent than the fastest known algorithm, given by Thilikos et al. (J Algorithms 56(1):1–24, 2005 ; J Algorithms 56(1):25–49, 2005 ), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.
AbstractList	Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2 O(k 3 log k). As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2 O(k 2 log k) · n, where k is the optimum width and n is the number of vertices. While being slower by a log k-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1-24, 2005] and [Cutwidth II: Algorithms for partial w-trees of bounded degree, J. Algorithms, 56(1):25-49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts. Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k . We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2 O ( k 3 log k ) . As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2 O ( k 2 log k ) · n , where k is the optimum width and n is the number of vertices. While being slower by a log k -factor in the exponent than the fastest known algorithm, given by Thilikos et al. (J Algorithms 56(1):1–24, 2005 ; J Algorithms 56(1):25–49, 2005 ), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts. Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2O(k3logk). As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2O(k2logk)·n, where k is the optimum width and n is the number of vertices. While being slower by a logk-factor in the exponent than the fastest known algorithm, given by Thilikos et al. (J Algorithms 56(1):1–24, 2005; J Algorithms 56(1):25–49, 2005), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.
Author	Giannopoulou, Archontia C. Thilikos, Dimitrios M. Wrochna, Marcin Pilipczuk, Michał Raymond, Jean-Florent
Author_xml	– sequence: 1 givenname: Archontia C. surname: Giannopoulou fullname: Giannopoulou, Archontia C. organization: Technische Universität Berlin – sequence: 2 givenname: Michał surname: Pilipczuk fullname: Pilipczuk, Michał organization: Institute of Informatics, University of Warsaw – sequence: 3 givenname: Jean-Florent surname: Raymond fullname: Raymond, Jean-Florent organization: Technische Universität Berlin – sequence: 4 givenname: Dimitrios M. surname: Thilikos fullname: Thilikos, Dimitrios M. organization: AlGCo Project Team, CNRS, LIRMM, Department of Mathematics, National and Kapodistrian University of Athens – sequence: 5 givenname: Marcin orcidid: 0000-0001-9346-2172 surname: Wrochna fullname: Wrochna, Marcin email: m.wrochna@mimuw.edu.pl organization: Institute of Informatics, University of Warsaw
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Keywords	Immersions Obstructions Cutwidth Fixed-parameter tractability cutwidth
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Combinatorial analysis Computer Science Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Design parameters Discrete Mathematics Graph theory Graphs Layouts Mathematics of Computing Obstructions Submerging Theory of Computation
Title	Cutwidth: Obstructions and Algorithmic Aspects
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