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Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications (2003)
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| Venue: | In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA’04 |
| Citations: | 31 - 10 self |
BibTeX
@INPROCEEDINGS{Demaine03equivalenceof,
author = {Erik D. Demaine and Mohammadtaghi Hajiaghayi},
title = {Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications},
booktitle = {In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA’04},
year = {2003},
pages = {833--842}
}
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Abstract
We solve an open problem posed by Eppstein in 1995 [14, 15] and re-enforced by Grohe [16, 17] concerning locally bounded treewidth in minor-closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minor-closed families of graphs with bounded local treewidth as precisely minor-closed families that minor-exclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apex-minor-free graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minor-closed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomial-time approximation schemes for a broad class of problems in apex-minor-free graphs, improving the running time from .
Keyphrases
local treewidth linear local treewidth minor-closed family algorithmic application apex-minor-free graph linear bound apex graph bounded local treewidth tighter bound several author planar graph open problem broad class exponential bound special class minor-closed graph family running time polynomial-time approximation scheme
