TY - JOUR
T1 - Cutwidth
T2 - Obstructions and Algorithmic Aspects
AU - Giannopoulou, Archontia C.
AU - Pilipczuk, Michał
AU - Raymond, Jean Florent
AU - Thilikos, Dimitrios M.
AU - Wrochna, Marcin
N1 - Publisher Copyright:
© 2018, The Author(s).
PY - 2019/2/15
Y1 - 2019/2/15
N2 - 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 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.
AB - 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 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.
KW - Cutwidth
KW - Fixed-parameter tractability
KW - Immersions
KW - Obstructions
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U2 - 10.1007/s00453-018-0424-7
DO - 10.1007/s00453-018-0424-7
M3 - Article
AN - SCOPUS:85044092082
SN - 0178-4617
VL - 81
SP - 557
EP - 588
JO - Algorithmica
JF - Algorithmica
IS - 2
ER -