Equivalence of local treewidth and linear local treewidth and its algorithmic applications

  • Authors:

  • Erik D. Demaine;MohammadTaghi Hajiaghayi

  • Affiliations:

  • Massachusetts Institute of Technology, Cambridge, MA;Massachusetts Institute of Technology, Cambridge, MA

  • Venue:
  • SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2004

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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 222O(1/ε) nO(1) to 2O(1/ε)nO(1).