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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Veröffentlicht in:Algorithmica Jg. 81; H. 2; S. 557 - 588
Hauptverfasser: Giannopoulou, Archontia C., Pilipczuk, Michał, Raymond, Jean-Florent, Thilikos, Dimitrios M., Wrochna, Marcin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.02.2019
Springer Nature B.V
Springer Verlag
Schlagworte:
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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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-018-0424-7