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treewidth - Computational Topology(Jeff Erickson Treewidth...

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Computational Topology (Jeff Erickson) Treewidth Or il y avait des graines terribles sur la planète du petit prince . . . c’étaient les graines de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils la font éclater. [Now there were some terrible seeds on the planet that was the home of the little prince; and these were the seeds of the baobab. The soil of that planet was infested with them. A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces.] — Antoine de Saint-Exupéry (translated by Katherine Woods) Le Petit Prince [The Little Prince] (1943) 11 Treewidth In this lecture, I will introduce a graph parameter called treewidth , that generalizes the property of having small separators. Intuitively, a graph has small treewidth if it can be recursively decomposed into small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’. Many problems that are NP-hard for general graphs can be solved in polynomial time for graphs with small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth, but they can be covered by a small number of subgraphs, each with small treewidth. This covering can be used to develop efficient algorithms (either exact or approximate) for a huge number of problems on surface graphs that are NP-hard for general graphs. As an example of this technique, I’ll describe a polynomial-time approximation scheme for the maximum independent set problem. 11.1 Definitions The concept of treewidth was discovered independently by several different researchers and given several different names. The actual term ‘treewidth’ and its definition in terms of tree decompositions were introduced by Robertson and Seymour [ 15 ] . Almost simultaneously, Arnborg and Proskurowski independently began the systematic study of partial k -trees [ 3 , 2 , 1 ] . Both groups were apparently unaware of Bertelè and Brioschi’s earlier equivalent definition of the dimension 1 of a graph [ 6 ] , or Halin’s related study of S -functions [ 13 ] . Other equivalent definitions are surveyed by Bodlaender [ 8 , 9 ] . A tree decomposition ( T , X ) of a graph G = ( V , E ) consists of a tree T = ( I , F ) and a function X : I 2 V satisfying three constraints. To avoid confusion, I will always refer to vertices of G , but nodes of T . We say that a vertex v is associated with a node i , or vice versa, whenever v X ( i ) .
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