Decomposition, approximation, and coloring of odd-minor-free graphs
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Demaine_Decomposition approximation.pdf
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Author(s)
Demaine, Erik D.
Hajiaghayi, Mohammad Taghi
Kawarabayashi, Ken-ichi
Date Issued
January 2010
Journal
ACM-SIAM Symposium on Discrete Algorithms
Publisher
Association for Computing Machinery / Society for Industrial and Applied Mathematics
Citation
Demaine, Erik D., Mohammad Taghi Hajiagharyi, Ken-ichi Kawarabayashi. "Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs" Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, Jan. 2010. 329-344. Copyright © 2010 Society for Industrial and Applied Mathematics.
Version
Author's final manuscript
Abstract
We prove two structural decomposition theorems about graphs excluding
a fixed odd minor H, and show how these theorems can
be used to obtain approximation algorithms for several algorithmic
problems in such graphs. Our decomposition results provide new
structural insights into odd-H-minor-free graphs, on the one hand
generalizing the central structural result from Graph Minor Theory,
and on the other hand providing an algorithmic decomposition
into two bounded-treewidth graphs, generalizing a similar result for
minors. As one example of how these structural results conquer difficult
problems, we obtain a polynomial-time 2-approximation for
vertex coloring in odd-H-minor-free graphs, improving on the previous
O(jV (H)j)-approximation for such graphs and generalizing
the previous 2-approximation for H-minor-free graphs. The class
of odd-H-minor-free graphs is a vast generalization of the well-studied
H-minor-free graph families and includes, for example, all
bipartite graphs plus a bounded number of apices. Odd-H-minor-free
graphs are particularly interesting from a structural graph theory
perspective because they break away from the sparsity of H-
minor-free graphs, permitting a quadratic number of edges.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://www.siam.org/proceedings/soda/2010/SODA10_028_demainee.pdf
