Local Tree-Width, Excluded Minors, and Approximation Algorithms

@article{Grohe2003LocalTE,
  title={Local Tree-Width, Excluded Minors,
and Approximation Algorithms},
  author={Martin Grohe},
  journal={Combinatorica},
  year={2003},
  volume={23},
  pages={613-632}
}
  • Martin Grohe
  • Published 2003
  • Mathematics, Computer Science
  • Combinatorica
The local tree-width of a graph G=(V,E) is the function ltwG :ℕ→ℕ that associates with every r∈ℕ the maximal tree-width of an r-neighborhood in G. Our main grapht heoretic result is a decomposition theorem for graphs with excluded minors, which says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width.As an application of this theorem, we show that a number of combinatorial optimization problems, suchas Minimum Vertex Cover, Minimum Dominating Set, and… Expand
Bidimensional Parameters and Local Treewidth
TLDR
It is proved that, for a large family of graph parameters called contraction-bidimensional, a minor-closed graph family has the parameter-treewidth property if $\mathcal{F}$ has bounded local treewidth. Expand
From Tree-Width to Clique-Width: Excluding a Unit Interval Graph
  • V. Lozin
  • Mathematics, Computer Science
  • ISAAC
  • 2008
TLDR
It is proved that the unit interval graphs constitute a minimal hereditary class of unbounded clique-width, and it is shown that list coloring is fixed parameter tractable in the class of unit intervals graphs. Expand
Fast approximation schemes for K3, 3-minor-free or K5-minor-free graphs
TLDR
It is proved that any graph excluding one of K_{5} or K_{3,3} as a minor has local treewidth bounded by 3k+4, which means that practical polynomial-time approximation schemes for both minimization and maximization problems on these classes of non-planar graphs can be designed. Expand
Bidimensional Parameters and Local Treewidth
TLDR
This paper examines the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far. Expand
Linearity of grid minors in treewidth with applications through bidimensionality
We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth,Expand
Diameter and Treewidth in Minor-Closed Graph Families, Revisited
TLDR
This short paper characterized the minor-closed graph families for which the treewidth is bounded by a function of the diameter with a simple proof of Eppstein's characterization, which includes, e.g., planar graphs. Expand
Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
TLDR
This paper compromises between generality and efficiency by considering the problem on graphs of small arboricity a, which includes, but is not limited, graphs excluding fixed minors, such as planar graphs, graphs of (locally) bounded treewidth, or bounded genus. Expand
Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality
We prove that any H-minor-free graph, for a fixed graph H, of treewidth ω has an Ω(ω) × Ω(ω) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth,Expand
Contraction decomposition in h-minor-free graphs and algorithmic applications
We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph ofExpand
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs
We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1). Our results apply to a broad family of graph problems, calledExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 34 REFERENCES
Diameter and Treewidth in Minor-Closed Graph Families
  • D. Eppstein
  • Mathematics, Computer Science
  • Algorithmica
  • 2000
TLDR
It is shown that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family, and the O(D) bound above can be extended to bounded-genus graphs. Expand
NC-Algorithms for Graphs with Small Treewidth
TLDR
A parallel algorithm for recognizing graphs with treewidth ≤ k, for constant k, and building the corresponding tree-decomposition, that uses O(log n) time and O(n3k+4) processors on a CRCW PRAM is given. Expand
A separator theorem for graphs with an excluded minor and its applications
TLDR
It follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and solve any sparse system of n linear equations in n unknowns in time O(n). Expand
Deciding First-Order Properties of Locally Tree-Decomposalbe Graphs
TLDR
It is shown that for each locally tree-decomposable class C of graphs and for each property φ of graphs that is definable in first-order logic, there is a linear time algorithm deciding whether a given graph G ∈ C has property ϕ. Expand
Easy Problems for Tree-Decomposable Graphs
Abstract Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets canExpand
Graph Minors. XVI. Excluding a non-planar graph
TLDR
It is found that every graph with no minor isomorphic to L may be constructed by piecing together in a tree-structure graphs each of which "almost" embeds in some surface in which L cannot be embedded. Expand
Graph minors. III. Planar tree-width
TLDR
It is proved that for any fixed planar graph H, every planar graphs with sufficiently large tree-width has a minor isomorphic to H. Expand
Approximation algorithms for NP-complete problems on planar graphs
  • B. Baker
  • Mathematics, Computer Science
  • 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
  • 1983
TLDR
A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand
A Separator Theorem for Planar Graphs
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains moreExpand
A polynomial-time approximation scheme for weighted planar graph TSP
TLDR
This work finds a salesman tour of total cost at most (1 + E) times optimal in time n for any E > 6, and presents a quasi-polynomial time algorithm for the Steiner version of this problem. Expand
...
1
2
3
4
...