Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications
Name
MIT-LCS-TR-903.pdf
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335.69 KB
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Adobe PDF
Checksum (MD5)
e262666e67f9808f700724afd70c125e
Author(s)
Demaine, Erik D.
Hajiaghayi, Mohammad Taghi
Date Issued
May 2003
Series/Report no.
MIT-LCS-TR-903
Abstract
We solve an open problem posed by Eppstein in 1995 and re-enforced by Grohe 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 2^(2^(2^O(1/epsilon))) n^O(1) to 2^O(1/epsilon) n^O(1).
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